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<title>Integer sequence</title>

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

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, an <b>integer sequence</b> is a <a href="Sequence" title="Sequence">sequence</a> (i.e., an ordered list) of <a href="Integer" title="Integer">integers</a>.

</p><p>An integer sequence may be specified <i>explicitly</i> by giving a formula for its <i>n</i>th term, or <i>implicitly</i> by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the <a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci sequence</a>) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description (sequence <span class="nowrap external"><a href="https://oeis.org/A000045" class="extiw external" title="oeis:A000045">A000045</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). The sequence 0, 3, 8, 15, ... is formed according to the formula <i>n</i><sup>2</sup> − 1 for the <i>n</i>th term: an explicit definition.

</p><p>Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a <a href="Perfect_number" title="Perfect number">perfect number</a>, (sequence <span class="nowrap external"><a href="https://oeis.org/A000396" class="extiw external" title="oeis:A000396">A000396</a></span> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>), even though we do not have a formula for the <i>n</i>th perfect number.

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<div class="mw-heading mw-heading2"><h2 id="Computable_and_definable_sequences">Computable and definable sequences</h2></div>

<p>An integer sequence is <b><a href="Computable_function" title="Computable function">computable</a></b> if there exists an algorithm that, given <i>n</i>, calculates <i>a</i><sub><i>n</i></sub>, for all <i>n</i> > 0. The set of computable integer sequences is <a href="Countable" class="mw-redirect" title="Countable">countable</a>. The set of all integer sequences is <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> (with <a href="Cardinality" title="Cardinality">cardinality</a> equal to <a href="Beth_one" class="mw-redirect" title="Beth one">that of the continuum</a>), and so not all integer sequences are computable.

</p><p>Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

</p><p>Suppose the set <i>M</i> is a <a href="Transitive_model" title="Transitive model">transitive model</a> of <a href="ZFC_set_theory" class="mw-redirect" title="ZFC set theory">ZFC set theory</a>. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a <b><a href="Definable_set" title="Definable set">definable</a> sequence relative to <i>M</i></b> if there exists some formula <i>P</i>(<i>x</i>) in the language of set theory, with one free variable and no parameters, which is true in <i>M</i> for that integer sequence and false in <i>M</i> for all other integer sequences. In each such <i>M</i>, there are definable integer sequences that are not computable, such as sequences that encode the <a href="Turing_jump" title="Turing jump">Turing jumps</a> of computable sets.

</p><p>For some transitive models <i>M</i> of ZFC, every sequence of integers in <i>M</i> is definable relative to <i>M</i>; for others, only some integer sequences are. There is no systematic way to define in <i>M</i> itself the set of sequences definable relative to <i>M</i> and that set may not even exist in some such <i>M</i>. Similarly, the map from the set of formulas that define integer sequences in <i>M</i> to the integer sequences they define is not definable in <i>M</i> and may not exist in <i>M</i>. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model.<sup id="cite_ref-FOOTNOTEHamkinsLinetskyReitz2013_1-0" class="reference"><a href="#cite_note-FOOTNOTEHamkinsLinetskyReitz2013-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>If <i>M</i> contains all integer sequences, then the set of integer sequences definable in <i>M</i> will exist in <i>M</i> and be countable and countable in <i>M</i>.

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<div class="mw-heading mw-heading2"><h2 id="Complete_sequences">Complete sequences</h2></div>

<p>A sequence of positive integers is called a <a href="Complete_sequence" title="Complete sequence">complete sequence</a> if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

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

<p>Integer sequences that have their own name include:

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

<li><a href="Baum%E2%80%93Sweet_sequence" title="Baum–Sweet sequence">Baum–Sweet sequence</a></li>

<li><a href="Bell_number" title="Bell number">Bell numbers</a></li>

<li><a href="Binomial_coefficient" title="Binomial coefficient">Binomial coefficients</a></li>

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

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

<li><a href="Composite_number" title="Composite number">Composite numbers</a></li>

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

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

<li><a href="Even_and_odd_numbers" class="mw-redirect" title="Even and odd numbers">Even and odd numbers</a></li>

<li><a href="Factorial" title="Factorial">Factorial</a> numbers</li>

<li><a href="Fibonacci_number" class="mw-redirect" title="Fibonacci number">Fibonacci numbers</a></li>

<li><a href="Fibonacci_word" title="Fibonacci word">Fibonacci word</a></li>

<li><a href="Figurate_numbers" class="mw-redirect" title="Figurate numbers">Figurate numbers</a></li>

<li><a href="Golomb_sequence" title="Golomb sequence">Golomb sequence</a></li>

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

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

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

<li><a href="Home_prime" title="Home prime">Home primes</a></li>

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

<li><a href="Juggler_sequence" title="Juggler sequence">Juggler sequence</a></li>

<li><a href="Kolakoski_sequence" title="Kolakoski sequence">Kolakoski sequence</a></li>

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

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

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

<li><a href="Natural_number" title="Natural number">Natural numbers</a></li>

<li><a href="Padovan_sequence" title="Padovan sequence">Padovan numbers</a></li>

<li><a href="Partition_number" class="mw-redirect" title="Partition number">Partition numbers</a></li>

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

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

<li><a href="Prime_number" title="Prime number">Prime numbers</a></li>

<li><a href="Pseudoprime" title="Pseudoprime">Pseudoprime</a> numbers</li>

<li><a href="Recam%C3%A1n's_sequence" title="Recamán's sequence">Recamán's sequence</a></li>

<li><a href="Regular_paperfolding_sequence" title="Regular paperfolding sequence">Regular paperfolding sequence</a></li>

<li><a href="Rudin%E2%80%93Shapiro_sequence" title="Rudin–Shapiro sequence">Rudin–Shapiro sequence</a></li>

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

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

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

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

<li><a href="Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a></li>

<li><a href="Ulam_numbers" class="mw-redirect" title="Ulam numbers">Ulam numbers</a></li>

<li><a href="Weird_number" title="Weird number">Weird numbers</a></li>

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

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<ul><li><a href="Constant-recursive_sequence" title="Constant-recursive sequence">Constant-recursive sequence</a></li>

<li><a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>

<ul><li><a href="List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">List of OEIS sequences</a></li></ul></li></ul>

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

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<li id="cite_note-FOOTNOTEHamkinsLinetskyReitz2013-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHamkinsLinetskyReitz2013_1-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHamkinsLinetskyReitz2013">Hamkins, Linetsky & Reitz 2013</a>.</span>

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</style><cite id="CITEREFHamkinsLinetskyReitz2013" class="citation cs2">Hamkins, Joel David; Linetsky, David; Reitz, Jonas (2013), "Pointwise Definable Models of Set Theory", <i><a href="Journal_of_Symbolic_Logic" title="Journal of Symbolic Logic">Journal of Symbolic Logic</a></i>, <b>78</b> (1): <span class="nowrap">139–</span>156, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1105.4597">1105.4597</a></span>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2178%2Fjsl.7801090">10.2178/jsl.7801090</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:43689192">43689192</a></cite>.</li></ul>

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

<ul><li><a rel="nofollow" class="external text" href="https://cs.uwaterloo.ca/journals/JIS/">Journal of Integer Sequences</a>. Articles are freely available online.</li></ul>

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</style><div id="Sequences_and_series332" style="font-size:114%;margin:0 4em"><a href="Sequence" title="Sequence">Sequences</a> and <a href="Series_(mathematics)" title="Series (mathematics)">series</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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%;text-align:left">Basic</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="Arithmetic_progression" title="Arithmetic progression">Arithmetic progression</a></li>

<li><a href="Geometric_progression" title="Geometric progression">Geometric progression</a></li>

<li><a href="Harmonic_progression_(mathematics)" title="Harmonic progression (mathematics)">Harmonic progression</a></li>

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

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

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

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

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

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

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Advanced <span class="nobold">(<a href="List_of_OEIS_sequences" class="mw-redirect" title="List of OEIS sequences">list</a>)</span></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="Complete_sequence" title="Complete sequence">Complete sequence</a></li>

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

<li><a href="Figurate_number" title="Figurate number">Figurate number</a></li>

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

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

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

<li><a href="Pell_number" title="Pell number">Pell number</a></li>

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

<li><a href="Polygonal_number" title="Polygonal number">Polygonal number</a></li>

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

<ul><li><a href="Triangular_array" title="Triangular array">array</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="Fibonacci_sequence" title="Fibonacci sequence"></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of sequences</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cauchy_sequence" title="Cauchy sequence">Cauchy sequence</a></li>

<li><a href="Monotonic_function" title="Monotonic function">Monotonic function</a></li>

<li><a href="Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Properties of series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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%;text-align:left">Series</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="Alternating_series" title="Alternating series">Alternating</a></li>

<li><a href="Convergent_series" title="Convergent series">Convergent</a></li>

<li><a href="Divergent_series" title="Divergent series">Divergent</a></li>

<li><a href="Telescoping_series" title="Telescoping series">Telescoping</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;text-align:left">Convergence</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="Absolute_convergence" title="Absolute convergence">Absolute</a></li>

<li><a href="Conditional_convergence" title="Conditional convergence">Conditional</a></li>

<li><a href="Uniform_convergence" title="Uniform convergence">Uniform</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Explicit series</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" 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%;text-align:left">Convergent</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Riemann_zeta_function" title="Riemann zeta function">1 + 1/2<sup><i>s</i></sup> + 1/3<sup><i>s</i></sup> + ... (Riemann zeta function)</a></li></ul>

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

<li><a href="Grandi's_series" title="Grandi's series">1 − 1 + 1 − 1 + ⋯ (Grandi's series)</a></li>

<li><a href="Infinite_arithmetic_series" class="mw-redirect" title="Infinite arithmetic series">Infinite arithmetic series</a></li>

<li><a href="1_%E2%88%92_1_%2B_2_%E2%88%92_6_%2B_24_%E2%88%92_120_%2B_%E2%8B%AF" title="1 − 1 + 2 − 6 + 24 − 120 + ⋯">1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)</a></li>

<li><a href="Harmonic_series_(mathematics)" title="Harmonic series (mathematics)">1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)</a></li>

<li><a href="Divergence_of_the_sum_of_the_reciprocals_of_the_primes" title="Divergence of the sum of the reciprocals of the primes">1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)</a></li></ul>

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

<ul><li><a href="Taylor_series" title="Taylor series">Taylor series</a></li>

<li><a href="Power_series" title="Power series">Power series</a></li>

<li><a href="Formal_power_series" title="Formal power series">Formal power series</a></li>

<li><a href="Laurent_series" title="Laurent series">Laurent series</a></li>

<li><a href="Puiseux_series" title="Puiseux series">Puiseux series</a></li>

<li><a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li>

<li><a href="Trigonometric_series" title="Trigonometric series">Trigonometric series</a></li>

<li><a href="Fourier_series" title="Fourier series">Fourier series</a></li>

<li><a href="Generating_series" class="mw-redirect" title="Generating series">Generating series</a></li></ul>

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

<ul><li><a href="Generalized_hypergeometric_series" class="mw-redirect" title="Generalized hypergeometric series">Generalized hypergeometric series</a></li>

<li><a href="Hypergeometric_function_of_a_matrix_argument" title="Hypergeometric function of a matrix argument">Hypergeometric function of a matrix argument</a></li>

<li><a href="Lauricella_hypergeometric_series" title="Lauricella hypergeometric series">Lauricella hypergeometric series</a></li>

<li><a href="Modular_hypergeometric_series" class="mw-redirect" title="Modular hypergeometric series">Modular hypergeometric series</a></li>

<li><a href="Riemann's_differential_equation" title="Riemann's differential equation">Riemann's differential equation</a></li>

<li><a href="Theta_hypergeometric_series" class="mw-redirect" title="Theta hypergeometric series">Theta hypergeometric series</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li></ul>

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