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<title>Padovan sequence</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Padovan sequence</span></span>

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<p>In <a href="Number_theory" title="Number theory">number theory</a>, the <b>Padovan sequence</b> is the <a href="Integer_sequence" title="Integer sequence">sequence of integers</a> <i>P</i>(<i>n</i>) defined<sup id="cite_ref-ps_1-0" class="reference"><a href="#cite_note-ps-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> by the initial values:

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(0)=P(1)=P(2)=1,}">

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<annotation encoding="application/x-tex">{\displaystyle P(0)=P(1)=P(2)=1,}</annotation>

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</p><p>and the <a href="Recurrence_relation" title="Recurrence relation">recurrence relation</a>

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)=P(n-2)+P(n-3).}">

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<annotation encoding="application/x-tex">{\displaystyle P(n)=P(n-2)+P(n-3).}</annotation>

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</p><p>The first few values of <i>P</i>(<i>n</i>) are

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<dl><dd>1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, ... (sequence <span class="nowrap external"><a href="https://oeis.org/A000931" class="extiw external" title="oeis:A000931">A000931</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl>

<p>The Padovan sequence is named after <a href="Richard_Padovan" title="Richard Padovan">Richard Padovan</a> who attributed its discovery to <a href="Netherlands" title="Netherlands">Dutch</a> architect <a href="Hans_van_der_Laan" title="Hans van der Laan">Hans van der Laan</a> in his 1994 essay <i>Dom. Hans van der Laan: Modern Primitive</i>.<sup id="cite_ref-dhdl_2-0" class="reference"><a href="#cite_note-dhdl-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The <a href="Sequence" title="Sequence">sequence</a> was described by <a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> in his <a href="Scientific_American" title="Scientific American">Scientific American</a> column <i>Mathematical Recreations</i> in June 1996.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> He also writes about it in one of his books, "Math Hysteria: Fun Games With Mathematics".

</p><p><i>The above definition is the one given by Ian Stewart and by <a href="MathWorld" title="MathWorld">MathWorld</a>. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.</i>

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<div class="mw-heading mw-heading2"><h2 id="Recurrence_relations">Recurrence relations</h2></div>

<p>In the spiral, each <a href="Triangle" title="Triangle">triangle</a> shares a side with two others giving a visual proof that

the Padovan sequence also satisfies the recurrence relation

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<annotation encoding="application/x-tex">{\displaystyle P(n)=P(n-1)+P(n-5)}</annotation>

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</math></span><img src="./bd5fceca8d5d2da6f1f4dd09f255053544e9176b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.793ex; height:2.843ex;" alt="{\displaystyle P(n)=P(n-1)+P(n-5)}" loading="lazy"></span></dd></dl>

<p>Starting from this, the defining recurrence and other recurrences as they are discovered,

one can create an infinite number of further recurrences by repeatedly replacing <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(m)}">

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<annotation encoding="application/x-tex">{\displaystyle P(m)}</annotation>

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</math></span><img src="./b3a233906e02f973dc3ce3d3fc3cacca780e3714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.595ex; height:2.843ex;" alt="{\displaystyle P(m)}" loading="lazy"></span> by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(m-2)+P(m-3)}">

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<annotation encoding="application/x-tex">{\displaystyle P(m-2)+P(m-3)}</annotation>

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</math></span><img src="./1f83ab4474ce1d6b7a8bb919782f092e18303e3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.036ex; height:2.843ex;" alt="{\displaystyle P(m-2)+P(m-3)}" loading="lazy"></span>

</p><p>The <a href="Perrin_pseudoprime" class="mw-redirect" title="Perrin pseudoprime">Perrin sequence</a> satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.

</p><p>The Perrin sequence can be obtained from the Padovan sequence by the

following formula:

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Perrin} (n)=P(n+1)+P(n-10).\,}</annotation>

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</math></span><img src="./906bb57c0ecfa6740f77a8a0621a4f708d3c745a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.622ex; height:2.843ex;" alt="{\displaystyle \mathrm {Perrin} (n)=P(n+1)+P(n-10).\,}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Extension_to_negative_parameters">Extension to negative parameters</h2></div>

<p>As with any sequence defined by a recurrence relation, Padovan numbers <i>P</i>(<i>m</i>) for <i>m</i><0 can be defined by rewriting the recurrence relation as

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<annotation encoding="application/x-tex">{\displaystyle P(m)=P(m+3)-P(m+1),}</annotation>

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</math></span><img src="./d412db6f600114b5397a4580fc2c6468627d3725.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.377ex; height:2.843ex;" alt="{\displaystyle P(m)=P(m+3)-P(m+1),}" loading="lazy"></span></dd></dl>

<p>Starting with <i>m</i> = −1 and working backwards, we extend <i>P</i>(<i>m</i>) to negative indices:

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<td>7

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<td>−7

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<td>4

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<td>0

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<td>−3

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<td>4

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<td>−3

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<td>1

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<td>1

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<td>−2

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<td>2

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<td>−1

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<td>0

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<td>1

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<td>−1

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<td>1

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<td>0

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<td>0

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<td>1

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<td>0

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<td>1

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<td>1

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<td>1

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<div class="mw-heading mw-heading2"><h2 id="Sums_of_terms">Sums of terms</h2></div>

<p>The sum of the first <i>n</i> terms in the Padovan sequence is 2 less than <i>P</i>(<i>n</i> + 5), i.e.

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(m)=P(n+5)-2.}</annotation>

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</math></span><img src="./08a1ebf670b87be18ed4211231581b54b4b02f9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.226ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(m)=P(n+5)-2.}" loading="lazy"></span></dd></dl>

<p>Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(2m)=P(2n+3)-1}</annotation>

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</math></span><img src="./6ba89107b7043093577898c48a6c93c7bbcc4ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.904ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(2m)=P(2n+3)-1}" loading="lazy"></span> <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A077855" class="extiw external" title="oeis:A077855">A077855</a></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(2m+1)=P(2n+4)-1}</annotation>

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</math></span><img src="./5a20bf710e79548d396029d2725e2d53adea9e19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.907ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(2m+1)=P(2n+4)-1}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(3m)=P(3n+2)}</annotation>

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</math></span><img src="./c559ad3cff5c0238c7547d3c16b6831d2070f753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.901ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(3m)=P(3n+2)}" loading="lazy"></span> <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A034943" class="extiw external" title="oeis:A034943">A034943</a></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(3m+1)=P(3n+3)-1}</annotation>

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</math></span><img src="./9c4f88eeb5aa75e6116d584ad480eba4423c7781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.907ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(3m+1)=P(3n+3)-1}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(3m+2)=P(3n+4)-1}</annotation>

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</math></span><img src="./8382689514c21c753020f0dd0d962f06128c24e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.907ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(3m+2)=P(3n+4)-1}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(5m)=P(5n+1).}</annotation>

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</math></span><img src="./b5eba257c8cc257ad4d82a95b02c5ab9f374a4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.548ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(5m)=P(5n+1).}" loading="lazy"></span> <span class="nowrap external"><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A012772" class="extiw external" title="oeis:A012772">A012772</a></span></dd></dl>

<p>Sums involving products of terms in the Padovan sequence satisfy the following identities:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(m)^{2}=P(n+2)^{2}-P(n-1)^{2}-P(n-3)^{2}}</annotation>

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</math></span><img src="./073311ad679ccdb95b02228261d4d5be84fe2e73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.379ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(m)^{2}=P(n+2)^{2}-P(n-1)^{2}-P(n-3)^{2}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(m)^{2}P(m+1)=P(n)P(n+1)P(n+2)}</annotation>

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</math></span><img src="./eb5ba248bb7701fa8c3ce5cad2109909a84ba5e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.13ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(m)^{2}P(m+1)=P(n)P(n+1)P(n+2)}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{m=0}^{n}P(m)P(m+2)=P(n+2)P(n+3)-1.}</annotation>

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</math></span><img src="./41502c60552472cd6529340f08b51e6b39778e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.776ex; height:6.843ex;" alt="{\displaystyle \sum _{m=0}^{n}P(m)P(m+2)=P(n+2)P(n+3)-1.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Other_identities">Other identities</h2></div>

<p>The Padovan sequence also satisfies the identity

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<annotation encoding="application/x-tex">{\displaystyle P(n)^{2}-P(n+1)P(n-1)=P(-n-7).\,}</annotation>

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</math></span><img src="./ef532c10015f183937a9f5f0fa7f2e25206e0abf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.641ex; height:3.176ex;" alt="{\displaystyle P(n)^{2}-P(n+1)P(n-1)=P(-n-7).\,}" loading="lazy"></span></dd></dl>

<p>The Padovan sequence is related to sums of <a href="Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a> by the following identity:

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<annotation encoding="application/x-tex">{\displaystyle P(k-2)=\sum _{2m+n=k}{m \choose n}=\sum _{m=\lceil k/3\rceil }^{\lfloor k/2\rfloor }{m \choose k-2m}.}</annotation>

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</math></span><img src="./dc2ca1cde687562249dd07417a2e636dafb7a888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:45.87ex; height:8.176ex;" alt="{\displaystyle P(k-2)=\sum _{2m+n=k}{m \choose n}=\sum _{m=\lceil k/3\rceil }^{\lfloor k/2\rfloor }{m \choose k-2m}.}" loading="lazy"></span></dd></dl>

<p>For example, for <i>k</i> = 12, the values for the pair (<i>m</i>, <i>n</i>) with 2<i>m</i> + <i>n</i> = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:

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<annotation encoding="application/x-tex">{\displaystyle {6 \choose 0}+{5 \choose 2}+{4 \choose 4}=1+10+1=12=P(10).\,}</annotation>

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</math></span><img src="./028083becd803291568dba646223abb4af4a4578.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:48.296ex; height:6.176ex;" alt="{\displaystyle {6 \choose 0}+{5 \choose 2}+{4 \choose 4}=1+10+1=12=P(10).\,}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Binet-like_formula">Binet-like formula</h2></div>

<p>The Padovan sequence numbers can be written in terms of powers of the <a href="Root_of_a_polynomial" class="mw-redirect" title="Root of a polynomial">roots</a> of the equation<sup id="cite_ref-ps_1-1" class="reference"><a href="#cite_note-ps-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle x^{3}-x-1=0.\,}</annotation>

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</math></span><img src="./55ef6db4c26c656f0137972ebf2889c52b96b442.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.852ex; height:2.843ex;" alt="{\displaystyle x^{3}-x-1=0.\,}" loading="lazy"></span></dd></dl>

<p>This equation has 3 roots; one <a href="Real_number" title="Real number">real</a> root <i>p</i> (known as the <a href="Plastic_ratio" title="Plastic ratio">plastic ratio</a>) and two <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a> roots <i>q</i> and <i>r</i>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Given these three roots, the Padovan sequence can be expressed by a formula involving <i>p</i>, <i>q</i> and <i>r</i> :

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<annotation encoding="application/x-tex">{\displaystyle P(n)=ap^{n}+bq^{n}+cr^{n}}</annotation>

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</math></span><img src="./694de47824e463028c68685972cf0ef9d8c8dee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.916ex; height:2.843ex;" alt="{\displaystyle P(n)=ap^{n}+bq^{n}+cr^{n}}" loading="lazy"></span></dd></dl>

<p>where <i>a</i>, <i>b</i> and <i>c</i> are constants.<sup id="cite_ref-ps_1-2" class="reference"><a href="#cite_note-ps-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>Since the <a href="Absolute_value" title="Absolute value">absolute values</a> of the <a href="Complex_number" title="Complex number">complex</a> roots <i>q</i> and <i>r</i> are both less than 1 (and hence <i>p</i> is a <a href="Pisot%E2%80%93Vijayaraghavan_number" title="Pisot–Vijayaraghavan number">Pisot–Vijayaraghavan number</a>), the powers of these roots <a href="Limit_of_a_sequence" title="Limit of a sequence">approach</a> 0 for large <i>n</i>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(n)-ap^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle P(n)-ap^{n}}</annotation>

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</math></span><img src="./0c06da7babb3602b88778dcc82b5b29955d7dcd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.408ex; height:2.843ex;" alt="{\displaystyle P(n)-ap^{n}}" loading="lazy"></span> tends to zero.

</p><p>For all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>

</semantics>

</math></span><img src="./ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}" loading="lazy"></span>, <i>P</i>(<i>n</i>) is the <a href="Nearest_integer" class="mw-redirect" title="Nearest integer">integer closest</a> to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p^{5}}{2p+3}}p^{n}}">

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {p^{5}}{2p+3}}p^{n}}</annotation>

</semantics>

</math></span><img src="./08510f62476b089ba885a0ac34b3118a96d827ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:9.559ex; height:6.176ex;" alt="{\displaystyle {\frac {p^{5}}{2p+3}}p^{n}}" loading="lazy"></span>. Indeed, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {p^{5}}{2p+3}}}">

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {p^{5}}{2p+3}}}</annotation>

</semantics>

</math></span><img src="./94a3e35fa0c4e36a7f3e1b26366190be745c48f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.171ex; height:6.176ex;" alt="{\displaystyle {\frac {p^{5}}{2p+3}}}" loading="lazy"></span> is the value of constant <i>a</i> above, while <i>b</i> and <i>c</i> are obtained by replacing <i>p</i> with <i>q</i> and <i>r</i>, respectively.

</p><p>The ratio of successive terms in the Padovan sequence approaches <i>p</i>, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the <a href="Perrin_sequence" class="mw-redirect" title="Perrin sequence">Perrin sequence</a> as the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> does to the <a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci sequence</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Combinatorial_interpretations">Combinatorial interpretations</h2></div>

<ul><li><i>P</i>(<i>n</i>) is the number of ways of writing <i>n</i> + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of <a href="Composition_(number_theory)" class="mw-redirect" title="Composition (number theory)">compositions</a> of <i>n</i> + 2 in which each term is either 2 or 3). For example, <i>P</i>(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:</li></ul>

<ul><li>The number of ways of writing <i>n</i> as an ordered sum in which no term is 2 is <i>P</i>(2<i>n</i> − 2). For example, <i>P</i>(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:</li></ul>

<ul><li>The number of ways of writing <i>n</i> as a palindromic ordered sum in which no term is 2 is <i>P</i>(<i>n</i>). For example, <i>P</i>(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:</li></ul>

<ul><li>The number of ways of writing <i>n</i> as an ordered sum in which each term is <a href="Parity_(mathematics)" title="Parity (mathematics)">odd</a> and greater than 1 is equal to <i>P</i>(<i>n</i> − 5). For example, <i>P</i>(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:</li></ul>

<ul><li>The number of ways of writing <i>n</i> as an ordered sum in which each term is <a href="Modular_arithmetic" title="Modular arithmetic">congruent</a> to 2 mod 3 is equal to <i>P</i>(<i>n</i> − 4). For example, <i>P</i>(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Generating_function">Generating function</h2></div>

<p>The <a href="Generating_function" title="Generating function">generating function</a> of the Padovan sequence is

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(P(n);x)={\frac {1+x}{1-x^{2}-x^{3}}}.}</annotation>

</semantics>

</math></span><img src="./a07241ed5f668aceb590ec925a6d4c98a37c3b85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.142ex; height:5.676ex;" alt="{\displaystyle G(P(n);x)={\frac {1+x}{1-x^{2}-x^{3}}}.}" loading="lazy"></span></dd></dl>

<p>This can be used to prove identities involving products of the Padovan sequence with <a href="Geometric_series" title="Geometric series">geometric terms</a>, such as:

</p>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {P(n)}{2^{n}}}={\frac {12}{5}}.}</annotation>

</semantics>

</math></span><img src="./3669f8c3c82e2d8b4e99d71e51c70e287ead2985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:16.434ex; height:6.843ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {P(n)}{2^{n}}}={\frac {12}{5}}.}" loading="lazy"></span></dd>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{n=0}^{\infty }{\frac {P(n)}{\alpha ^{n}}}={\frac {\alpha ^{2}(\alpha +1)}{\alpha ^{3}-\alpha -1}}.}</annotation>

</semantics>

</math></span><img src="./d554f35917a815c950afe185ea6eb9c3e9cd0424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.982ex; height:7.009ex;" alt="{\displaystyle \sum _{n=0}^{\infty }{\frac {P(n)}{\alpha ^{n}}}={\frac {\alpha ^{2}(\alpha +1)}{\alpha ^{3}-\alpha -1}}.}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<p>In a similar way to the <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a> that can be generalized to a set of <a href="Polynomial" title="Polynomial">polynomials</a>

called the <a href="Fibonacci_polynomials" title="Fibonacci polynomials">Fibonacci polynomials</a>, the Padovan sequence numbers can be generalized to

yield the <a href="Padovan_polynomials" title="Padovan polynomials">Padovan polynomials</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Padovan_L-system">Padovan L-system</h2></div>

<p>If we define the following simple grammar:

</p>

<dl><dd><b>variables</b> : A B C</dd>

<dd><b>constants</b> : none</dd>

<dd><b>start</b>  : A</dd>

<dd><b>rules</b>  : (A → B), (B → C), (C → AB)</dd></dl>

<p>then this Lindenmayer system or <a href="L-system" title="L-system">L-system</a> produces the following sequence of strings:

</p>

<dd><i>n</i> = 7 : BCCAB</dd>

<dd><i>n</i> = 8 : CABABBC</dd></dl>

<p>and if we count the length of each string, we obtain the Padovan numbers:

</p>

<p>Also, if you count the number of <i>A</i>s, <i>B</i>s and <i>C</i>s in each string, then for the <i>n</i>th

string, you have <i>P</i>(<i>n</i> − 5) <i>A</i>s, <i>P</i>(<i>n</i> − 3) <i>B</i>s and <i>P</i>(<i>n</i> − 4) <i>C</i>s. The count of <i>BB</i> pairs

and <i>CC</i> pairs are also Padovan numbers.

</p>

<div class="mw-heading mw-heading2"><h2 id="Cuboid_spiral">Cuboid spiral</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Padovan_cuboid_spiral" title="Padovan cuboid spiral">Padovan cuboid spiral</a></div>

<p>A spiral can be formed based on connecting the corners of a set of 3-dimensional <a href="Cuboid" title="Cuboid">cuboids</a>.

This is the <a href="Padovan_cuboid_spiral" title="Padovan cuboid spiral">Padovan cuboid spiral</a>. Successive sides of this spiral have lengths that are

the Padovan numbers multiplied by the <a href="Square_root_of_2" title="Square root of 2">square root of 2</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Pascal's_triangle">Pascal's triangle</h2></div>

<p><a href="Erv_Wilson" title="Erv Wilson">Erv Wilson</a> in his paper <i>The Scales of Mt. Meru</i><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> observed certain diagonals in <a href="Pascal's_triangle" title="Pascal's triangle">Pascal's triangle</a> (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) observed that these diagonals generate the Padovan sequence by summing the diagonal numbers.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p><span typeof="mw:File"></span>

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PadovanSequence.html">"Padovan Sequence"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span>.</span>

</li>

<li id="cite_note-dhdl-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-dhdl_2-0">^</a></b></span> <span class="reference-text">Richard Padovan. <i>Dom Hans van der Laan: modern primitive</i>: Architectura & Natura Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9789071570407</bdi>.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Ian Stewart, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060131123213/http://members.fortunecity.com/templarser/padovan.html"><i>Tales of a Neglected Number</i></a>, <i>Scientific American</i>, No. 6, June 1996, pp. 92-93.</span>

</li>

<li id="cite_note-stewart-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-stewart_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFIan_Stewart2004" class="citation cs2"><a href="Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> (2004), <i>Math hysteria: fun and games with mathematics</i>, Oxford University Press, p. 87, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-861336-7</bdi></cite>.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Richard Padovan, <a rel="nofollow" class="external text" href="http://www.nexusjournal.com/conferences/N2002-Padovan.html">"Dom Hans Van Der Laan and the Plastic Number"</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201226025734/http://www.nexusjournal.com/conferences/N2002-Padovan.html">Archived</a> 2020-12-26 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, pp. 181-193 in Nexus IV: Architecture and Mathematics, eds. <a href="Kim_Williams_(architect)" title="Kim Williams (architect)">Kim Williams</a> and Jose Francisco Rodrigues, Fucecchio (Florence): Kim Williams Books, 2002.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Erv Wilson (1993), <a rel="nofollow" class="external text" href="http://www.anaphoria.com/meruone.pdf"><i>Scales of Mt. Meru</i></a></span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFSloane_"A000931"" class="citation web cs1"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A000931">"Sequence A000931"</a>. <i>The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite> See formula credited to Paul Barry, July 6, 2004</span>

</li>

</ol></div></div>

<ul><li>Ian Stewart, A Guide to Computer Dating (Feedback), Scientific American, Vol. 275, No. 5, November 1996, Pg. 118.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><abbr title="On-Line Encyclopedia of Integer Sequences">OEIS</abbr> <a rel="nofollow" class="external text" href="https://oeis.org/A000931">sequence A000931 (Padovan sequence)</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070216024906/http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp">A Padovan sequence calculator</a></li></ul>

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</style><div id="Classes_of_natural_numbers743" style="font-size:114%;margin:0 4em">Classes of <a href="Natural_number" title="Natural number">natural numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Powers_and_related_numbers743" style="font-size:114%;margin:0 4em"><a href="Exponentiation" title="Exponentiation">Powers</a> and related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Power_of_two" title="Power of two">Power of 2</a></li>

<li><a href="Power_of_three" title="Power of three">Power of 3</a></li>

<li><a href="Power_of_10" title="Power of 10">Power of 10</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Fourth_power" title="Fourth power">Fourth power</a></li>

<li><a href="Fifth_power_(algebra)" title="Fifth power (algebra)">Fifth power</a></li>

<li><a href="Sixth_power" title="Sixth power">Sixth power</a></li>

<li><a href="Seventh_power" title="Seventh power">Seventh power</a></li>

<li><a href="Eighth_power" title="Eighth power">Eighth power</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Prime_power" title="Prime power">Prime power</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Of_the_form_a_×_2b_±_1743" style="font-size:114%;margin:0 4em">Of the form <i>a</i> × 2<sup><i>b</i></sup> ± 1</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cullen_number" title="Cullen number">Cullen</a></li>

<li><a href="Double_Mersenne_number" title="Double Mersenne number">Double Mersenne</a></li>

<li><a href="Fermat_number" title="Fermat number">Fermat</a></li>

<li><a href="Mersenne_prime" title="Mersenne prime">Mersenne</a></li>

<li><a href="Proth_number" class="mw-redirect" title="Proth number">Proth</a></li>

<li><a href="Thabit_number" title="Thabit number">Thabit</a></li>

<li><a href="Woodall_number" title="Woodall number">Woodall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_polynomial_numbers743" style="font-size:114%;margin:0 4em">Other polynomial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hilbert_number" title="Hilbert number">Hilbert</a></li>

<li><a href="Idoneal_number" title="Idoneal number">Idoneal</a></li>

<li><a href="Leyland_number" title="Leyland number">Leyland</a></li>

<li><a href="Loeschian_number" class="mw-redirect" title="Loeschian number">Loeschian</a></li>

<li><a href="Lucky_numbers_of_Euler" title="Lucky numbers of Euler">Lucky numbers of Euler</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Recursively_defined_numbers743" style="font-size:114%;margin:0 4em"><a href="Recursion" title="Recursion">Recursively</a> defined numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Fibonacci_sequence" title="Fibonacci sequence">Fibonacci</a></li>

<li><a href="Jacobsthal_number" title="Jacobsthal number">Jacobsthal</a></li>

<li><a href="Leonardo_number" title="Leonardo number">Leonardo</a></li>

<li><a href="Lucas_number" title="Lucas number">Lucas</a></li>

<li><a href="Supergolden_ratio#Narayana_sequence" title="Supergolden ratio">Narayana</a></li>

<li><a href="Perrin_number" title="Perrin number">Perrin</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Possessing_a_specific_set_of_other_numbers743" style="font-size:114%;margin:0 4em">Possessing a specific set of other numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amenable_number" title="Amenable number">Amenable</a></li>

<li><a href="Congruent_number" title="Congruent number">Congruent</a></li>

<li><a href="Kn%C3%B6del_number" title="Knödel number">Knödel</a></li>

<li><a href="Riesel_number" title="Riesel number">Riesel</a></li>

<li><a href="Sierpi%C5%84ski_number" title="Sierpiński number">Sierpiński</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Expressible_via_specific_sums743" style="font-size:114%;margin:0 4em">Expressible via specific sums</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonhypotenuse_number" title="Nonhypotenuse number">Nonhypotenuse</a></li>

<li><a href="Polite_number" title="Polite number">Polite</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primary_pseudoperfect_number" title="Primary pseudoperfect number">Primary pseudoperfect</a></li>

<li><a href="Wolstenholme_number" title="Wolstenholme number">Wolstenholme</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Figurate_numbers743" style="font-size:114%;margin:0 4em"><a href="Figurate_number" title="Figurate number">Figurate numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Plane_(mathematics)" title="Plane (mathematics)">2-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polygonal_number" title="Centered polygonal number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_triangular_number" title="Centered triangular number">Centered triangular</a></li>

<li><a href="Centered_square_number" title="Centered square number">Centered square</a></li>

<li><a href="Centered_pentagonal_number" title="Centered pentagonal number">Centered pentagonal</a></li>

<li><a href="Centered_hexagonal_number" title="Centered hexagonal number">Centered hexagonal</a></li>

<li><a href="Centered_heptagonal_number" title="Centered heptagonal number">Centered heptagonal</a></li>

<li><a href="Centered_octagonal_number" title="Centered octagonal number">Centered octagonal</a></li>

<li><a href="Centered_nonagonal_number" title="Centered nonagonal number">Centered nonagonal</a></li>

<li><a href="Centered_decagonal_number" title="Centered decagonal number">Centered decagonal</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polygonal_number" title="Polygonal number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Triangular_number" title="Triangular number">Triangular</a></li>

<li><a href="Square_number" title="Square number">Square</a></li>

<li><a href="Square_triangular_number" title="Square triangular number">Square triangular</a></li>

<li><a href="Pentagonal_number" title="Pentagonal number">Pentagonal</a></li>

<li><a href="Hexagonal_number" title="Hexagonal number">Hexagonal</a></li>

<li><a href="Heptagonal_number" title="Heptagonal number">Heptagonal</a></li>

<li><a href="Octagonal_number" title="Octagonal number">Octagonal</a></li>

<li><a href="Nonagonal_number" title="Nonagonal number">Nonagonal</a></li>

<li><a href="Decagonal_number" title="Decagonal number">Decagonal</a></li>

<li><a href="Dodecagonal_number" title="Dodecagonal number">Dodecagonal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Three-dimensional_space" title="Three-dimensional space">3-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Centered_polyhedral_number" title="Centered polyhedral number">centered</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Centered_tetrahedral_number" title="Centered tetrahedral number">Centered tetrahedral</a></li>

<li><a href="Centered_cube_number" title="Centered cube number">Centered cube</a></li>

<li><a href="Centered_octahedral_number" title="Centered octahedral number">Centered octahedral</a></li>

<li><a href="Centered_dodecahedral_number" title="Centered dodecahedral number">Centered dodecahedral</a></li>

<li><a href="Centered_icosahedral_number" title="Centered icosahedral number">Centered icosahedral</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Polyhedral_number" class="mw-redirect" title="Polyhedral number">non-centered</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tetrahedral_number" title="Tetrahedral number">Tetrahedral</a></li>

<li><a href="Cube_(algebra)" title="Cube (algebra)">Cubic</a></li>

<li><a href="Octahedral_number" title="Octahedral number">Octahedral</a></li>

<li><a href="Dodecahedral_number" title="Dodecahedral number">Dodecahedral</a></li>

<li><a href="Icosahedral_number" title="Icosahedral number">Icosahedral</a></li>

<li><a href="Stella_octangula_number" title="Stella octangula number">Stella octangula</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Pyramidal_number" title="Pyramidal number">pyramidal</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Square_pyramidal_number" title="Square pyramidal number">Square pyramidal</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Four-dimensional_space" title="Four-dimensional space">4-dimensional</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">non-centered</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pentatope_number" title="Pentatope number">Pentatope</a></li>

<li><a href="Squared_triangular_number" title="Squared triangular number">Squared triangular</a></li>

<li><a href="Fourth_power" title="Fourth power">Tesseractic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_numbers743" style="font-size:114%;margin:0 4em">Combinatorial numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Catalan_number" title="Catalan number">Catalan</a></li>

<li><a href="Dedekind_number" title="Dedekind number">Dedekind</a></li>

<li><a href="Delannoy_number" title="Delannoy number">Delannoy</a></li>

<li><a href="Euler_number" class="mw-redirect" title="Euler number">Euler</a></li>

<li><a href="Eulerian_number" title="Eulerian number">Eulerian</a></li>

<li><a href="Fuss%E2%80%93Catalan_number" title="Fuss–Catalan number">Fuss–Catalan</a></li>

<li><a href="Lazy_caterer's_sequence" title="Lazy caterer's sequence">Lazy caterer's sequence</a></li>

<li><a href="Motzkin_number" title="Motzkin number">Motzkin</a></li>

<li><a href="Narayana_number" title="Narayana number">Narayana</a></li>

<li><a href="Ordered_Bell_number" title="Ordered Bell number">Ordered Bell</a></li>

<li><a href="Schr%C3%B6der_number" title="Schröder number">Schröder</a></li>

<li><a href="Schr%C3%B6der%E2%80%93Hipparchus_number" title="Schröder–Hipparchus number">Schröder–Hipparchus</a></li>

<li><a href="Stirling_numbers_of_the_first_kind" title="Stirling numbers of the first kind">Stirling first</a></li>

<li><a href="Stirling_numbers_of_the_second_kind" title="Stirling numbers of the second kind">Stirling second</a></li>

<li><a href="Telephone_number_(mathematics)" title="Telephone number (mathematics)">Telephone number</a></li>

<li><a href="Wedderburn%E2%80%93Etherington_number" title="Wedderburn–Etherington number">Wedderburn–Etherington</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Primes743" style="font-size:114%;margin:0 4em"><a href="Prime_number" title="Prime number">Primes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Wieferich_prime#Wieferich_numbers" title="Wieferich prime">Wieferich</a></li>

<li><a href="Wolstenholme_prime" title="Wolstenholme prime">Wolstenholme prime</a></li>

<li><a href="Wilson_prime#Wilson_numbers" title="Wilson prime">Wilson</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Pseudoprimes743" style="font-size:114%;margin:0 4em"><a href="Pseudoprime" title="Pseudoprime">Pseudoprimes</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carmichael_number" title="Carmichael number">Carmichael number</a></li>

<li><a href="Catalan_pseudoprime" title="Catalan pseudoprime">Catalan pseudoprime</a></li>

<li><a href="Elliptic_pseudoprime" title="Elliptic pseudoprime">Elliptic pseudoprime</a></li>

<li><a href="Euler_pseudoprime" title="Euler pseudoprime">Euler pseudoprime</a></li>

<li><a href="Euler%E2%80%93Jacobi_pseudoprime" title="Euler–Jacobi pseudoprime">Euler–Jacobi pseudoprime</a></li>

<li><a href="Fermat_pseudoprime" title="Fermat pseudoprime">Fermat pseudoprime</a></li>

<li><a href="Frobenius_pseudoprime" title="Frobenius pseudoprime">Frobenius pseudoprime</a></li>

<li><a href="Lucas_pseudoprime" title="Lucas pseudoprime">Lucas pseudoprime</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael number</a></li>

<li><a href="Perrin_number#Perrin_primality_test" title="Perrin number">Perrin pseudoprime</a></li>

<li><a href="Somer%E2%80%93Lucas_pseudoprime" title="Somer–Lucas pseudoprime">Somer–Lucas pseudoprime</a></li>

<li><a href="Strong_pseudoprime" title="Strong pseudoprime">Strong pseudoprime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Arithmetic_functions_and_dynamics743" style="font-size:114%;margin:0 4em"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Divisor_function" title="Divisor function">Divisor functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Prime_omega_function" title="Prime omega function">Prime omega functions</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Almost_prime" title="Almost prime">Almost prime</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Highly_cototient_number" title="Highly cototient number">Highly cototient</a></li>

<li><a href="Highly_totient_number" title="Highly totient number">Highly totient</a></li>

<li><a href="Noncototient" title="Noncototient">Noncototient</a></li>

<li><a href="Nontotient" title="Nontotient">Nontotient</a></li>

<li><a href="Perfect_totient_number" title="Perfect totient number">Perfect totient</a></li>

<li><a href="Sparsely_totient_number" title="Sparsely totient number">Sparsely totient</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequences</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a></li>

<li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Sociable_numbers" class="mw-redirect" title="Sociable numbers">Sociable</a></li>

<li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Primorial" title="Primorial">Primorial</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Euclid_number" title="Euclid number">Euclid</a></li>

<li><a href="Fortunate_number" title="Fortunate number">Fortunate</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Other_prime_factor_or_divisor_related_numbers743" style="font-size:114%;margin:0 4em">Other <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factor</a> or <a href="Divisor" title="Divisor">divisor</a> related numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Cyclic_number_(group_theory)" title="Cyclic number (group theory)">Cyclic</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li>

<li><a href="Erd%C5%91s%E2%80%93Woods_number" title="Erdős–Woods number">Erdős–Woods</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Giuga_number" title="Giuga number">Giuga</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Jordan%E2%80%93P%C3%B3lya_number" title="Jordan–Pólya number">Jordan–Pólya</a></li>

<li><a href="Lucas%E2%80%93Carmichael_number" title="Lucas–Carmichael number">Lucas–Carmichael</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="St%C3%B8rmer_number" title="Størmer number">Størmer</a></li>

<li><a href="Super-Poulet_number" title="Super-Poulet number">Super-Poulet</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Numeral_system-dependent_numbers743" style="font-size:114%;margin:0 4em"><a href="Numeral_system" title="Numeral system">Numeral system</a>-dependent numbers</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Arithmetic_function" title="Arithmetic function">Arithmetic functions</a> <br>and <a href="Arithmetic_dynamics" title="Arithmetic dynamics">dynamics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Persistence_of_a_number" title="Persistence of a number">Persistence</a>

<ul><li><a href="Additive_persistence" class="mw-redirect" title="Additive persistence">Additive</a></li>

<li><a href="Multiplicative_persistence" class="mw-redirect" title="Multiplicative persistence">Multiplicative</a></li></ul></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Digit_sum" title="Digit sum">Digit sum</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Digit_sum" title="Digit sum">Digit sum</a></li>

<li><a href="Digital_root" title="Digital root">Digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit product</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Multiplicative_digital_root" title="Multiplicative digital root">Multiplicative digital root</a></li>

<li><a href="Sum-product_number" title="Sum-product number">Sum-product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Coding-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Meertens_number" title="Meertens number">Meertens</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dudeney_number" title="Dudeney number">Dudeney</a></li>

<li><a href="Factorion" title="Factorion">Factorion</a></li>

<li><a href="Kaprekar_number" title="Kaprekar number">Kaprekar</a></li>

<li><a href="Kaprekar's_routine" title="Kaprekar's routine">Kaprekar's constant</a></li>

<li><a href="Keith_number" title="Keith number">Keith</a></li>

<li><a href="Lychrel_number" title="Lychrel number">Lychrel</a></li>

<li><a href="Narcissistic_number" title="Narcissistic number">Narcissistic</a></li>

<li><a href="Perfect_digit-to-digit_invariant" title="Perfect digit-to-digit invariant">Perfect digit-to-digit invariant</a></li>

<li><a href="Perfect_digital_invariant" title="Perfect digital invariant">Perfect digital invariant</a>

<ul><li><a href="Happy_number" title="Happy number">Happy</a></li></ul></li></ul>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="P-adic_numbers" class="mw-redirect" title="P-adic numbers">P-adic numbers</a>-related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Automorphic_number" title="Automorphic number">Automorphic</a>

<ul><li><a href="Trimorphic_number" class="mw-redirect" title="Trimorphic number">Trimorphic</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_digit" title="Numerical digit">Digit</a>-composition related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Palindromic_number" title="Palindromic number">Palindromic</a></li>

<li><a href="Pandigital_number" title="Pandigital number">Pandigital</a></li>

<li><a href="Repdigit" title="Repdigit">Repdigit</a></li>

<li><a href="Repunit" title="Repunit">Repunit</a></li>

<li><a href="Self-descriptive_number" title="Self-descriptive number">Self-descriptive</a></li>

<li><a href="Smarandache%E2%80%93Wellin_number" title="Smarandache–Wellin number">Smarandache–Wellin</a></li>

<li><a href="Undulating_number" title="Undulating number">Undulating</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Digit-<a href="Permutation" title="Permutation">permutation</a> related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cyclic_number" title="Cyclic number">Cyclic</a></li>

<li><a href="Digit-reassembly_number" title="Digit-reassembly number">Digit-reassembly</a></li>

<li><a href="Parasitic_number" title="Parasitic number">Parasitic</a></li>

<li><a href="Primeval_number" title="Primeval number">Primeval</a></li>

<li><a href="Transposable_integer" title="Transposable integer">Transposable</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Divisor-related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li>

<li><a href="Vampire_number" title="Vampire number">Vampire</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Friedman_number" title="Friedman number">Friedman</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Binary_numbers743" style="font-size:114%;margin:0 4em"><a href="Binary_number" title="Binary number">Binary numbers</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Odious_number" title="Odious number">Odious</a></li>

<li><a href="Pernicious_number" title="Pernicious number">Pernicious</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Generated_via_a_sieve743" style="font-size:114%;margin:0 4em">Generated via a <a href="Sieve_theory" title="Sieve theory">sieve</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Lucky_number" title="Lucky number">Lucky</a></li>

<li><a href="Generation_of_primes" title="Generation of primes">Prime</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Sorting_related743" style="font-size:114%;margin:0 4em"><a href="Sorting_algorithm" title="Sorting algorithm">Sorting</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Pancake_sorting" title="Pancake sorting">Pancake number</a></li>

<li><a href="Sorting_number" title="Sorting number">Sorting number</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Natural_language_related743" style="font-size:114%;margin:0 4em"><a href="Natural_language" title="Natural language">Natural language</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aronson's_sequence" title="Aronson's sequence">Aronson's sequence</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Graphemics_related743" style="font-size:114%;margin:0 4em"><a href="Graphemics" title="Graphemics">Graphemics</a> related</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Strobogrammatic_number" title="Strobogrammatic number">Strobogrammatic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="2" style="font-weight:bold;"><div>

<ul><li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2025-07-21" href="https://en.wikipedia.org/wiki/?title=Padovan_sequence&oldid=1301776496">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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