This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A000027 M0472 N0173 #694 May 26 2025 09:56:04
%S A000027 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A000027 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U A000027 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77
%N A000027 The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.
%C A000027 For some authors, the terms "natural numbers" and "counting numbers" include 0, i.e., refer to the nonnegative integers A001477; the term "whole numbers" frequently also designates the whole set of (signed) integers A001057.
%C A000027 a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
%C A000027 Inverse Euler transform of A000219.
%C A000027 The rectangular array having A000027 as antidiagonals is the dispersion of the complement of the triangular numbers, A000217 (which triangularly form column 1 of this array). The array is also the transpose of A038722. - _Clark Kimberling_, Apr 05 2003
%C A000027 For nonzero x, define f(n) = floor(nx) - floor(n/x). Then f=A000027 if and only if x=tau or x=-tau. - _Clark Kimberling_, Jan 09 2005
%C A000027 Numbers of form (2^i)*k for odd k (i.e., n = A006519(n)*A000265(n)); thus n corresponds uniquely to an ordered pair (i,k) where i=A007814, k=A000265 (with A007814(2n)=A001511(n), A007814(2n+1)=0). - _Lekraj Beedassy_, Apr 22 2006
%C A000027 If the offset were changed to 0, we would have the following pattern: a(n)=binomial(n,0) + binomial(n,1) for the present sequence (number of regions in 1-space defined by n points), A000124 (number of regions in 2-space defined by n straight lines), A000125 (number of regions in 3-space defined by n planes), A000127 (number of regions in 4-space defined by n hyperplanes), A006261, A008859, A008860, A008861, A008862 and A008863, where the last six sequences are interpreted analogously and in each "... by n ..." clause an offset of 0 has been assumed, resulting in a(0)=1 for all of them, which corresponds to the case of not cutting with a hyperplane at all and therefore having one region. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
%C A000027 Define a number of points on a straight line to be in general arrangement when no two points coincide. Then these are the numbers of regions defined by n points in general arrangement on a straight line, when an offset of 0 is assumed. For instance, a(0)=1, since using no point at all leaves one region. The sequence satisfies the recursion a(n) = a(n-1) + 1. This has the following geometrical interpretation: Suppose there are already n-1 points in general arrangement, thus defining the maximal number of regions on a straight line obtainable by n-1 points, and now one more point is added in general arrangement. Then it will coincide with no other point and act as a dividing wall thereby creating one new region in addition to the a(n-1)=(n-1)+1=n regions already there, hence a(n)=a(n-1)+1. Cf. the comments on A000124 for an analogous interpretation. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 19 2006
%C A000027 The sequence a(n)=n (for n=1,2,3) and a(n)=n+1 (for n=4,5,...) gives to the rank (minimal cardinality of a generating set) for the semigroup I_n\S_n, where I_n and S_n denote the symmetric inverse semigroup and symmetric group on [n]. - _James East_, May 03 2007
%C A000027 The sequence a(n)=n (for n=1,2), a(n)=n+1 (for n=3) and a(n)=n+2 (for n=4,5,...) gives the rank (minimal cardinality of a generating set) for the semigroup PT_n\T_n, where PT_n and T_n denote the partial transformation semigroup and transformation semigroup on [n]. - _James East_, May 03 2007
%C A000027 "God made the integers; all else is the work of man." This famous quotation is a translation of "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk," spoken by Leopold Kronecker in a lecture at the Berliner Naturforscher-Versammlung in 1886. Possibly the first publication of the statement is in Heinrich Weber's "Leopold Kronecker," Jahresberichte D.M.V. 2 (1893) 5-31. - _Clark Kimberling_, Jul 07 2007
%C A000027 Binomial transform of A019590, inverse binomial transform of A001792. - _Philippe Deléham_, Oct 24 2007
%C A000027 Writing A000027 as N, perhaps the simplest one-to-one correspondence between N X N and N is this: f(m,n) = ((m+n)^2 - m - 3n + 2)/2. Its inverse is given by I(k)=(g,h), where g = k - J(J-1)/2, h = J + 1 - g, J = floor((1 + sqrt(8k - 7))/2). Thus I(1)=(1,1), I(2)=(1,2), I(3)=(2,1) and so on; the mapping I fills the first-quadrant lattice by successive antidiagonals. - _Clark Kimberling_, Sep 11 2008
%C A000027 a(n) is also the mean of the first n odd integers. - _Ian Kent_, Dec 23 2008
%C A000027 Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - _Gary W. Adamson_, Jun 05 2009
%C A000027 These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - _Michael B. Porter_, Oct 08 2009
%C A000027 Totally multiplicative sequence with a(p) = p for prime p. Totally multiplicative sequence with a(p) = a(p-1) + 1 for prime p. - _Jaroslav Krizek_, Oct 18 2009
%C A000027 Triangle T(k,j) of natural numbers, read by rows, with T(k,j) = binomial(k,2) + j = (k^2-k)/2 + j where 1 <= j <= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) < n and j = n - binomial(k,2). For example, T(4,1)=7, T(4,2)=8, T(4,3)=9, and T(4,4)=10. Note that T(n,n)=A000217(n), the n-th triangular number. - _Dennis P. Walsh_, Nov 19 2009
%C A000027 Hofstadter-Conway-like sequence (see A004001): a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = 1, a(2) = 2. - _Jaroslav Krizek_, Dec 11 2009
%C A000027 a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - _Leonid Bedratyuk_, Jan 04 2010
%C A000027 Floyd's triangle read by rows. - _Paul Muljadi_, Jan 25 2010
%C A000027 Number of numbers between k and 2k where k is an integer. - _Giovanni Teofilatto_, Mar 26 2010
%C A000027 Generated from a(2n) = r*a(n), a(2n+1) = a(n) + a(n+1), r = 2; in an infinite set, row 2 of the array shown in A178568. - _Gary W. Adamson_, May 29 2010
%C A000027 1/n = continued fraction [n]. Let barover[n] = [n,n,n,...] = 1/k. Then k - 1/k = n. Example: [2,2,2,...] = (sqrt(2) - 1) = 1/k, with k = (sqrt(2) + 1). Then 2 = k - 1/k. - _Gary W. Adamson_, Jul 15 2010
%C A000027 Number of n-digit numbers the binary expansion of which contains one run of 1's. - _Vladimir Shevelev_, Jul 30 2010
%C A000027 From _Clark Kimberling_, Jan 29 2011: (Start)
%C A000027 Let T denote the "natural number array A000027":
%C A000027 1 2 4 7 ...
%C A000027 3 5 8 12 ...
%C A000027 6 9 13 18 ...
%C A000027 10 14 19 25 ...
%C A000027 T(n,k) = n+(n+k-2)*(n+k-1)/2. See A185787 for a list of sequences based on T, such as rows, columns, diagonals, and sub-arrays. (End)
%C A000027 The Stern polynomial B(n,x) evaluated at x=2. See A125184. - _T. D. Noe_, Feb 28 2011
%C A000027 The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - _Mohammad K. Azarian_, Oct 13 2011
%C A000027 As a function of Bernoulli numbers B_n (cf. A027641: (1, -1/2, 1/6, 0, -1/30, 0, 1/42, ...)): let V = a variant of B_n changing the (-1/2) to (1/2). Then triangle A074909 (the beheaded Pascal's triangle) * [1, 1/2, 1/6, 0, -1/30, ...] = the vector [1, 2, 3, 4, 5, ...]. - _Gary W. Adamson_, Mar 05 2012
%C A000027 Number of partitions of 2n+1 into exactly two parts. - _Wesley Ivan Hurt_, Jul 15 2013
%C A000027 Integers n dividing u(n) = 2u(n-1) - u(n-2); u(0)=0, u(1)=1 (Lucas sequence A001477). - _Thomas M. Bridge_, Nov 03 2013
%C A000027 For this sequence, the generalized continued fraction a(1)+a(1)/(a(2)+a(2)/(a(3)+a(3)/(a(4)+...))), evaluates to 1/(e-2) = A194807. - _Stanislav Sykora_, Jan 20 2014
%C A000027 Engel expansion of e-1 (A091131 = 1.71828...). - _Jaroslav Krizek_, Jan 23 2014
%C A000027 a(n) is the number of permutations of length n simultaneously avoiding 213, 231 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - _Manda Riehl_, Aug 05 2014
%C A000027 a(n) is also the number of permutations simultaneously avoiding 213, 231 and 321 in the classical sense which can be realized as labels on an increasing strict binary tree with 2n-1 nodes. See A245904 for more information on increasing strict binary trees. - _Manda Riehl_, Aug 07 2014
%C A000027 a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) < 1/n. - _Clark Kimberling_, Sep 28 2014
%C A000027 a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. - _Clark Kimberling_, Oct 02 2014
%C A000027 Determinants of the spiral knots S(2,k,(1)). a(k) = det(S(2,k,(1))). These knots are also the torus knots T(2,k). - _Ryan Stees_, Dec 15 2014
%C A000027 As a function, the restriction of the identity map on the nonnegative integers {0,1,2,3...}, A001477, to the positive integers {1,2,3,...}. - _M. F. Hasler_, Jan 18 2015
%C A000027 See also A131685(k) = smallest positive number m such that c(i) = m (i^1 + 1) (i^2 + 2) ... (i^k+ k) / k! takes integral values for all i>=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - _Alexander R. Povolotsky_, Apr 24 2015
%C A000027 a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016
%C A000027 Does not satisfy Benford's law [Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017
%C A000027 Parametrization for the finite multisubsets of the positive integers, where, for p_j the j-th prime, n = Product_{j} p_j^(e_j) corresponds to the multiset containing e_j copies of j ('Heinz encoding' -- see A056239, A003963, A289506, A289507, A289508, A289509). - _Christopher J. Smyth_, Jul 31 2017
%C A000027 The arithmetic function v_1(n,1) as defined in A289197. - _Robert Price_, Aug 22 2017
%C A000027 For n >= 3, a(n)=n is the least area that can be obtained for an irregular octagon drawn in a square of n units side, whose sides are parallel to the axes, with 4 vertices that coincide with the 4 vertices of the square, and the 4 remaining vertices having integer coordinates. See Affaire de Logique link. - _Michel Marcus_, Apr 28 2018
%C A000027 a(n+1) is the order of rowmotion on a poset defined by a disjoint union of chains of length n. - _Nick Mayers_, Jun 08 2018
%C A000027 Number of 1's in n-th generation of 1-D Cellular Automata using Rules 50, 58, 114, 122, 178, 186, 206, 220, 238, 242, 250 or 252 in the Wolfram numbering scheme, started with a single 1. - _Frank Hollstein_, Mar 25 2019
%C A000027 (1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - _Gary W. Adamson_, Jul 15 2019
%D A000027 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
%D A000027 T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
%D A000027 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 22.
%D A000027 W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From _Leonid Bedratyuk_, Jan 04 2010]
%D A000027 I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
%D A000027 R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
%D A000027 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000027 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000027 N. J. A. Sloane, <a href="/A000027/b000027.txt">Table of n, a(n) for n = 1..500000</a> [a large file]
%H A000027 Archimedes Laboratory, <a href="http://www.archimedes-lab.org/numbers/Num1_200.html">What's special about this number?</a>
%H A000027 Affaire de Logique, <a href="http://www.affairedelogique.com/espace_probleme.php?corps=probleme&num=1051">Pick et Pick et Colegram</a> (in French), No. 1051, 18-04-2018.
%H A000027 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A000027 James Barton, <a href="http://www.virtuescience.com/number-list.html">The Numbers</a>
%H A000027 A. Berger and T. P. Hill, <a href="http://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf">What is Benford's Law?</a>, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.
%H A000027 A. Breiland, L. Oesper, and L. Taalman, <a href="http://educ.jmu.edu/~taalmala/breil_oesp_taal.pdf">p-Coloring classes of torus knots</a>, Online Missouri J. Math. Sci., 21 (2009), 120-126.
%H A000027 N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, <a href="http://projecteuclid.org/euclid.mjms/1312232716">Spiral knots</a>, Missouri J. of Math. Sci., 22 (2010).
%H A000027 C. K. Caldwell, <a href="https://t5k.org/curios">Prime Curios</a>
%H A000027 Case and Abiessu, <a href="http://everything2.net/index.pl?node_id=17633&displaytype=printable&lastnode_id=17633">interesting number</a>
%H A000027 S. Crandall, <a href="http://tingilinde.typepad.com/starstuff/2005/11/significant_int.html">notes on interesting digital ephemera</a>
%H A000027 O. Curtis, <a href="http://users.pipeline.com.au/owen/Numbers.html">Interesting Numbers</a>
%H A000027 M. DeLong, M. Russell, and J. Schrock, <a href="http://dx.doi.org/10.2140/involve.2015.8.361">Colorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)</a>, Involve, Vol. 8 (2015), No. 3, 361-384.
%H A000027 Walter Felscher, <a href="http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may99/0210.html">Historia Matematica Mailing List Archive.</a>
%H A000027 P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 371.
%H A000027 Robert R. Forslund, <a href="http://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps">A logical alternative to the existing positional number system</a>, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.
%H A000027 E. Friedman, <a href="https://erich-friedman.github.io/numbers.html">What's Special About This Number?</a>
%H A000027 R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H A000027 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H A000027 Kival Ngaokrajang, <a href="/A000027/a000027_2.pdf">Illustration about relation to many other sequences</a>, when the sequence is considered as a triangular table read by its antidiagonals. <a href="/A000027/a000027_3.pdf">Additional illustrations</a> when the sequence is considered as a centered triangular table read by rows.
%H A000027 Mike Keith, <a href="https://web.archive.org/web/20080509124530/http://users.aol.com/s6sj7gt/interest.htm">All Numbers Are Interesting: A Constructive Approach</a>
%H A000027 Leonardo of Pisa [Leonardo Pisano], <a href="/A000027/a000027.jpg">Illustration of initial terms</a>, from Liber Abaci [The Book of Calculation], 1202 (photo by David Singmaster).
%H A000027 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See pp. 15, 24.
%H A000027 Robert Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>.
%H A000027 G. Pfeiffer, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html">Counting Transitive Relations</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
%H A000027 R. Phillips, <a href="http://richardphillips.org.uk/number/Num1.htm">Numbers from one to thirty-one</a>
%H A000027 J. Striker, <a href="http://www.ams.org/publications/journals/notices/201706/rnoti-p543.pdf">Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance</a>, Notices of the AMS, June/July 2017, pp. 543-549.
%H A000027 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/aNombre/Nb0a50.htm">NOMBRES en BREF (in French)</a>
%H A000027 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NaturalNumber.html">Natural Number</a>, <a href="https://mathworld.wolfram.com/PositiveInteger.html">Positive Integer</a>, <a href="https://mathworld.wolfram.com/CountingNumber.html">Counting Number</a> <a href="https://mathworld.wolfram.com/Composition.html">Composition</a>, <a href="https://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Davenport-Schinzel Sequence</a>, <a href="https://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>, <a href="https://mathworld.wolfram.com/N.html">N</a>, <a href="https://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>, <a href="https://mathworld.wolfram.com/WholeNumber.html">Whole Number</a>, <a href="https://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>, and <a href="https://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H A000027 Wikipedia, <a href="http://en.wikipedia.org/wiki/List_of_numbers">List of numbers</a>, <a href="http://en.wikipedia.org/wiki/Interesting_number_paradox">Interesting number paradox</a>, and <a href="http://en.wikipedia.org/wiki/Floyd%27s_triangle">Floyd's triangle</a>
%H A000027 Robert G. Wilson v, <a href="/A000027/a000027.txt">English names for the numbers from 0 to 11159 without spaces or hyphens</a>
%H A000027 Robert G. Wilson v, <a href="/A001477/a001477.txt">American English names for the numbers from 0 to 100999 without spaces or hyphens</a>
%H A000027 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H A000027 <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%H A000027 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H A000027 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H A000027 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%H A000027 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A000027 <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F A000027 a(2k+1) = A005408(k), k >= 0, a(2k) = A005843(k), k >= 1.
%F A000027 Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001
%F A000027 Another g.f.: Sum_{n>0} phi(n)*x^n/(1-x^n) (Apostol).
%F A000027 When seen as an array: T(k, n) = n+1 + (k+n)*(k+n+1)/2. Main diagonal is 2n*(n+1)+1 (A001844), antidiagonal sums are n*(n^2+1)/2 (A006003). - _Ralf Stephan_, Oct 17 2004
%F A000027 Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005
%F A000027 G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
%F A000027 Series reversion of g.f. A(x) is x*C(-x)^2 where C(x) is the g.f. of A000108. - _Michael Somos_, Sep 04 2006
%F A000027 G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 4*u*v. - _Michael Somos_, Oct 03 2006
%F A000027 Convolution of A000012 (the all-ones sequence) with itself. - _Tanya Khovanova_, Jun 22 2007
%F A000027 a(n) = 2*a(n-1)-a(n-2); a(1)=1, a(2)=2. a(n) = 1+a(n-1). - _Philippe Deléham_, Nov 03 2008
%F A000027 a(n) = A000720(A000040(n)). - _Juri-Stepan Gerasimov_, Nov 29 2009
%F A000027 a(n+1) = Sum_{k=0..n} A101950(n,k). - _Philippe Deléham_, Feb 10 2012
%F A000027 a(n) = Sum_{d | n} phi(d) = Sum_{d | n} A000010(d). - _Jaroslav Krizek_, Apr 20 2012
%F A000027 G.f.: x * Product_{j>=0} (1+x^(2^j))^2 = x * (1+2*x+x^2) * (1+2*x^2+x^4) * (1+2*x^4+x^8) * ... = x + 2x^2 + 3x^3 + ... . - _Gary W. Adamson_, Jun 26 2012
%F A000027 a(n) = det(binomial(i+1,j), 1 <= i,j <= n). - _Mircea Merca_, Apr 06 2013
%F A000027 E.g.f.: x*E(0), where E(k) = 1 + 1/(x - x^3/(x^2 + (k+1)/E(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 03 2013
%F A000027 From _Wolfdieter Lang_, Oct 09 2013: (Start)
%F A000027 a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n > 1.
%F A000027 a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 1.
%F A000027 These identities are used in the calculation of products of ratios of lengths of certain lines in a regular n-gon. For the first identity see the Gradstein-Ryshik reference, p. 62, 1.392 1., bringing the first factor there to the left hand side and taking the limit x -> 0 (L'Hôpital). The second line follows from the first one. Thanks to _Seppo Mustonen_ who led me to consider n-gon lengths products. (End)
%F A000027 a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k>=0. - _Mircea Merca_, Jan 25 2014
%F A000027 a(n) = A052410(n)^A052409(n). - _Reinhard Zumkeller_, Apr 06 2014
%F A000027 a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - _Pierre CAMI_, Apr 25 2014
%F A000027 a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - _Clark Kimberling_, Oct 08 2014
%F A000027 a(n) = floor(1/(log(n+1)-log(n))). - _Thomas Ordowski_, Oct 10 2014
%F A000027 a(k) = det(S(2,k,1)). - _Ryan Stees_, Dec 15 2014
%F A000027 a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + ...). - _Pierre CAMI_, Jan 22 2015
%F A000027 a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n >= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m >= 0, from the fact that Stirling2*Stirling1 = identity matrix. See A048993, A048994 and A000110. - _Wolfdieter Lang_, Feb 03 2015
%F A000027 a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k*(2n-k). In addition, surprisingly, a(n) = Sum_{k=1..2n-1}(-1)^(k+1)*k^2*(2n-k)^2. - _Charlie Marion_, Jan 05 2016
%F A000027 G.f.: x/(1-x)^2 = (x * r(x) *r(x^3) * r(x^9) * r(x^27) * ...), where r(x) = (1 + x + x^2)^2 = (1 + 2x + 3x^2 + 2x^3 + x^4). - _Gary W. Adamson_, Jan 11 2017
%F A000027 a(n) = floor(1/(Pi/2-arctan(n))). - _Clark Kimberling_, Mar 11 2020
%F A000027 a(n) = Sum_{d|n} mu(n/d)*sigma(d). - _Ridouane Oudra_, Oct 03 2020
%F A000027 a(n) = Sum_{k=1..n} phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 09 2021
%F A000027 a(n) = S(n-1, 2), with the Chebyshev S-polynomials A049310. - _Wolfdieter Lang_, Mar 09 2023
%F A000027 From _Peter Bala_, Nov 02 2024: (Start)
%F A000027 For positive integer m, a(n) = (1/m)* Sum_{k = 1..2*m*n-1} (-1)^(k+1) * k * (2*m*n - k) = (1/m) * Sum_{k = 1..2*m*n-1} (-1)^(k+1) * k^2 * (2*m*n - k)^2 (the case m = 1 is given above).
%F A000027 a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * k * binomial(3*n+k, 2*k). (End)
%p A000027 A000027 := n->n; seq(A000027(n), n=1..100);
%t A000027 Range@ 77 (* _Robert G. Wilson v_, Mar 31 2015 *)
%o A000027 (Magma) [ n : n in [1..100]];
%o A000027 (PARI) {a(n) = n};
%o A000027 (R) 1:100
%o A000027 (Shell) seq 1 100
%o A000027 (Haskell)
%o A000027 a000027 = id
%o A000027 a000027_list = [1..] -- _Reinhard Zumkeller_, May 07 2012
%o A000027 (Maxima) makelist(n, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */
%o A000027 (Python)
%o A000027 def A000027(n): return n # _Chai Wah Wu_, May 09 2022
%o A000027 (Julia) print([n for n in 1:280]) # _Paul Muljadi_, Apr 09 2024
%o A000027 (Perl) print join(", ", 1..280) # _Paul Muljadi_, May 29 2024
%Y A000027 A001477 = nonnegative numbers.
%Y A000027 Partial sums of A000012.
%Y A000027 Cf. A001478, A001906, A007931, A007932, A027641, A074909, A089353 (multisets), A178568, A194807.
%Y A000027 Cf. A026081 = integers in reverse alphabetical order in U.S. English, A107322 = English name for number and its reverse have the same number of letters, A119796 = zero through ten in alphabetical order of English reverse spelling, A005589, etc. Cf. A185787 (includes a list of sequences based on the natural number array A000027).
%Y A000027 Cf. Boustrophedon transforms: A000737, A231179;
%Y A000027 Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).
%Y A000027 Cf. A048993, A048994, A000110 (see the Feb 03 2015 formula).
%Y A000027 Cf. A289187, A000010, A008683, A000203, A049310.
%K A000027 core,nonn,easy,mult,tabl
%O A000027 1,2
%A A000027 _N. J. A. Sloane_
%E A000027 Links edited by _Daniel Forgues_, Oct 07 2009.