oeis2-1.0.10: test/fixtures/A000027.json
[
{
"number": 27,
"id": "M0472 N0173",
"data": "1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77",
"name": "The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.",
"comment": [
"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.",
"a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).",
"Inverse Euler transform of A000219.",
"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",
"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",
"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",
"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",
"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",
"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",
"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",
"\"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",
"Binomial transform of A019590, inverse binomial transform of A001792. - _Philippe Deléham_, Oct 24 2007",
"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",
"a(n) is also the mean of the first n odd integers. - _Ian Kent_, Dec 23 2008",
"Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - _Gary W. Adamson_, Jun 05 2009",
"These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - _Michael B. Porter_, Oct 08 2009",
"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",
"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 \u003c= j \u003c= k. In other words, a(n) = n = binomial(k,2) + j where k is the largest integer such that binomial(k,2) \u003c 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",
"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",
"a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - _Leonid Bedratyuk_, Jan 04 2010",
"Floyd's triangle read by rows. - _Paul Muljadi_, Jan 25 2010",
"Number of numbers between k and 2k where k is an integer. - _Giovanni Teofilatto_, Mar 26 2010",
"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",
"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",
"Number of n-digit numbers the binary expansion of which contains one run of 1's. - _Vladimir Shevelev_, Jul 30 2010",
"From _Clark Kimberling_, Jan 29 2011: (Start)",
"Let T denote the \"natural number array A000027\":",
" 1 2 4 7 ...",
" 3 5 8 12 ...",
" 6 9 13 18 ...",
" 10 14 19 25 ...",
"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)",
"The Stern polynomial B(n,x) evaluated at x=2. See A125184. - _T. D. Noe_, Feb 28 2011",
"The denominator in the Maclaurin series of log(2), which is 1 - 1/2 + 1/3 - 1/4 + .... - _Mohammad K. Azarian_, Oct 13 2011",
"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",
"Number of partitions of 2n+1 into exactly two parts. - _Wesley Ivan Hurt_, Jul 15 2013",
"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",
"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",
"Engel expansion of e-1 (A091131 = 1.71828...). - _Jaroslav Krizek_, Jan 23 2014",
"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",
"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",
"a(n) = least k such that 2*Pi - Sum_{h=1..k} 1/(h^2 - h + 3/16) \u003c 1/n. - _Clark Kimberling_, Sep 28 2014",
"a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 \u003c 1/n. - _Clark Kimberling_, Oct 02 2014",
"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",
"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",
"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\u003e=0: For k=1, A131685(k)=1, which implies that this is a well defined integer sequence. - _Alexander R. Povolotsky_, Apr 24 2015",
"a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016",
"Does not satisfy Benford's law [Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017",
"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",
"The arithmetic function v_1(n,1) as defined in A289197. - _Robert Price_, Aug 22 2017",
"For n \u003e= 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",
"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",
"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",
"(1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - _Gary W. Adamson_, Jul 15 2019",
"a(n) is the chromatic number and the clique number of the complete graph K_n. - _Igor Blokhin_, Mar 09 2026"
],
"reference": [
"T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.",
"T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.",
"John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 22.",
"W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From _Leonid Bedratyuk_, Jan 04 2010]",
"I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.",
"R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.",
"N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).",
"N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence)."
],
"link": [
"N. J. A. Sloane, \u003ca href=\"/A000027/b000027.txt\"\u003eTable of n, a(n) for n = 1..500000\u003c/a\u003e [a large file]",
"Archimedes Laboratory, \u003ca href=\"http://www.archimedes-lab.org/numbers/Num1_200.html\"\u003eWhat's special about this number?\u003c/a\u003e",
"Affaire de Logique, \u003ca href=\"http://www.affairedelogique.com/espace_probleme.php?corps=probleme\u0026amp;num=1051\"\u003ePick et Pick et Colegram\u003c/a\u003e (in French), No. 1051, 18-04-2018.",
"Paul Barry, \u003ca href=\"http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html\"\u003eA Catalan Transform and Related Transformations on Integer Sequences\u003c/a\u003e, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.",
"James Barton, \u003ca href=\"http://www.virtuescience.com/number-list.html\"\u003eThe Numbers\u003c/a\u003e",
"A. Berger and T. P. Hill, \u003ca href=\"http://www.ams.org/publications/journals/notices/201702/rnoti-p132.pdf\"\u003eWhat is Benford's Law?\u003c/a\u003e, Notices, Amer. Math. Soc., 64:2 (2017), 132-134.",
"A. Breiland, L. Oesper, and L. Taalman, \u003ca href=\"https://doi.org/10.35834/mjms/1316027244\"\u003ep-Coloring classes of torus knots\u003c/a\u003e, Online Missouri J. Math. Sci., 21 (2009), 120-126.",
"N. Brothers, S. Evans, L. Taalman, L. Van Wyk, D. Witczak, and C. Yarnall, \u003ca href=\"https://doi.org/10.35834/mjms/1312232716\"\u003eSpiral knots\u003c/a\u003e, Missouri J. of Math. Sci., 22 (2010).",
"C. K. Caldwell, \u003ca href=\"https://t5k.org/curios\"\u003ePrime Curios\u003c/a\u003e",
"Case and Abiessu, \u003ca href=\"http://everything2.net/index.pl?node_id=17633\u0026amp;displaytype=printable\u0026amp;lastnode_id=17633\"\u003einteresting number\u003c/a\u003e",
"S. Crandall, \u003ca href=\"http://tingilinde.typepad.com/starstuff/2005/11/significant_int.html\"\u003enotes on interesting digital ephemera\u003c/a\u003e",
"O. Curtis, \u003ca href=\"http://users.pipeline.com.au/owen/Numbers.html\"\u003eInteresting Numbers\u003c/a\u003e",
"M. DeLong, M. Russell, and J. Schrock, \u003ca href=\"https://doi.org/10.2140/involve.2015.8.361\"\u003eColorability and determinants of T(m,n,r,s) twisted torus knots for n equiv. +/-1(mod m)\u003c/a\u003e, Involve, Vol. 8 (2015), No. 3, 361-384.",
"Walter Felscher, \u003ca href=\"http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may99/0210.html\"\u003eHistoria Matematica Mailing List Archive.\u003c/a\u003e",
"P. Flajolet and R. Sedgewick, \u003ca href=\"http://algo.inria.fr/flajolet/Publications/books.html\"\u003eAnalytic Combinatorics\u003c/a\u003e, 2009; see page 371.",
"Robert R. Forslund, \u003ca href=\"https://www.emis.de/journals/SWJPAM/Vol1_1995/rrf01.ps\"\u003eA logical alternative to the existing positional number system\u003c/a\u003e, Southwest Journal of Pure and Applied Mathematics, Vol. 1 1995 pp. 27-29.",
"E. Friedman, \u003ca href=\"https://erich-friedman.github.io/numbers.html\"\u003eWhat's Special About This Number?\u003c/a\u003e",
"R. K. Guy, \u003ca href=\"/A000346/a000346.pdf\"\u003eLetter to N. J. A. Sloane\u003c/a\u003e",
"Milan Janjic, \u003ca href=\"https://old.pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf\"\u003eEnumerative Formulas for Some Functions on Finite Sets\u003c/a\u003e",
"Mike Keith, \u003ca href=\"https://web.archive.org/web/20080509124530/http://users.aol.com/s6sj7gt/interest.htm\"\u003eAll Numbers Are Interesting: A Constructive Approach\u003c/a\u003e",
"Leonardo of Pisa [Leonardo Pisano], \u003ca href=\"/A000027/a000027.jpg\"\u003eIllustration of initial terms\u003c/a\u003e, from Liber Abaci [The Book of Calculation], 1202 (photo by David Singmaster).",
"Feihu Liu, Guoce Xin, and Chen Zhang, \u003ca href=\"https://arxiv.org/abs/2412.18744\"\u003eEhrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS\u003c/a\u003e, arXiv:2412.18744 [math.CO], 2024. See pp. 15, 24.",
"Robert Munafo, \u003ca href=\"http://www.mrob.com/pub/math/numbers.html\"\u003eNotable Properties of Specific Numbers\u003c/a\u003e.",
"Kival Ngaokrajang, \u003ca href=\"/A000027/a000027_2.pdf\"\u003eIllustration about relation to many other sequences\u003c/a\u003e, when the sequence is considered as a triangular table read by its antidiagonals. \u003ca href=\"/A000027/a000027_3.pdf\"\u003eAdditional illustrations\u003c/a\u003e when the sequence is considered as a centered triangular table read by rows.",
"G. Pfeiffer, \u003ca href=\"http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Pfeiffer/pfeiffer6.html\"\u003eCounting Transitive Relations\u003c/a\u003e, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.",
"R. Phillips, \u003ca href=\"http://richardphillips.org.uk/number/Num1.htm\"\u003eNumbers from one to thirty-one\u003c/a\u003e",
"J. Striker, \u003ca href=\"http://www.ams.org/publications/journals/notices/201706/rnoti-p543.pdf\"\u003eDynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance\u003c/a\u003e, Notices of the AMS, June/July 2017, pp. 543-549.",
"G. Villemin's Almanac of Numbers, \u003ca href=\"http://villemin.gerard.free.fr/aNombre/Nb0a50.htm\"\u003eNOMBRES en BREF (in French)\u003c/a\u003e",
"Eric Weisstein's World of Mathematics, \u003ca href=\"https://mathworld.wolfram.com/NaturalNumber.html\"\u003eNatural Number\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/PositiveInteger.html\"\u003ePositive Integer\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/CountingNumber.html\"\u003eCounting Number\u003c/a\u003e \u003ca href=\"https://mathworld.wolfram.com/Composition.html\"\u003eComposition\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/Davenport-SchinzelSequence.html\"\u003eDavenport-Schinzel Sequence\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/IdempotentNumber.html\"\u003eIdempotent Number\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/N.html\"\u003eN\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/SmarandacheCeilFunction.html\"\u003eSmarandache Ceil Function\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/WholeNumber.html\"\u003eWhole Number\u003c/a\u003e, \u003ca href=\"https://mathworld.wolfram.com/EngelExpansion.html\"\u003eEngel Expansion\u003c/a\u003e, and \u003ca href=\"https://mathworld.wolfram.com/TrinomialCoefficient.html\"\u003eTrinomial Coefficient\u003c/a\u003e",
"Wikipedia, \u003ca href=\"https://en.wikipedia.org/wiki/List_of_numbers\"\u003eList of numbers\u003c/a\u003e, \u003ca href=\"https://en.wikipedia.org/wiki/Interesting_number_paradox\"\u003eInteresting number paradox\u003c/a\u003e, and \u003ca href=\"https://en.wikipedia.org/wiki/Floyd%27s_triangle\"\u003eFloyd's triangle\u003c/a\u003e",
"Robert G. Wilson v, \u003ca href=\"/A000027/a000027.txt\"\u003eEnglish names for the numbers from 0 to 11159 without spaces or hyphens\u003c/a\u003e",
"Robert G. Wilson v, \u003ca href=\"/A001477/a001477.txt\"\u003eAmerican English names for the numbers from 0 to 100999 without spaces or hyphens\u003c/a\u003e",
"\u003ca href=\"/index/Cor#core\"\u003eIndex entries for \"core\" sequences\u003c/a\u003e",
"\u003ca href=\"/index/Aa#aan\"\u003eIndex entries for sequences of the a(a(n)) = 2n family\u003c/a\u003e",
"\u003ca href=\"/index/Per#IntegerPermutation\"\u003eIndex entries for sequences that are permutations of the natural numbers\u003c/a\u003e",
"\u003ca href=\"/index/Par#partN\"\u003eIndex entries for related partition-counting sequences\u003c/a\u003e",
"\u003ca href=\"/index/Rec#order_02\"\u003eIndex entries for linear recurrences with constant coefficients\u003c/a\u003e, signature (2,-1).",
"\u003ca href=\"/index/Di#divseq\"\u003eIndex to divisibility sequences\u003c/a\u003e",
"\u003ca href=\"/index/Be#Benford\"\u003eIndex entries for sequences related to Benford's law\u003c/a\u003e"
],
"formula": [
"a(2k+1) = A005408(k), k \u003e= 0, a(2k) = A005843(k), k \u003e= 1.",
"Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001",
"Another g.f.: Sum_{n\u003e0} phi(n)*x^n/(1-x^n) (Apostol).",
"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",
"Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005",
"G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).",
"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",
"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",
"Convolution of A000012 (the all-ones sequence) with itself. - _Tanya Khovanova_, Jun 22 2007",
"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",
"a(n) = A000720(A000040(n)). - _Juri-Stepan Gerasimov_, Nov 29 2009",
"a(n+1) = Sum_{k=0..n} A101950(n,k). - _Philippe Deléham_, Feb 10 2012",
"a(n) = Sum_{d | n} phi(d) = Sum_{d | n} A000010(d). - _Jaroslav Krizek_, Apr 20 2012",
"G.f.: x * Product_{j\u003e=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",
"a(n) = det(binomial(i+1,j), 1 \u003c= i,j \u003c= n). - _Mircea Merca_, Apr 06 2013",
"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",
"From _Wolfdieter Lang_, Oct 09 2013: (Start)",
"a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n \u003e 1.",
"a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n \u003e 1.",
"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 -\u003e 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)",
"a(n) = Sum_{j=0..k} (-1)^(j-1)*j*binomial(n,j)*binomial(n-1+k-j,k-j), k\u003e=0. - _Mircea Merca_, Jan 25 2014",
"a(n) = A052410(n)^A052409(n). - _Reinhard Zumkeller_, Apr 06 2014",
"a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - _Pierre CAMI_, Apr 25 2014",
"a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - _Clark Kimberling_, Oct 08 2014",
"a(n) = floor(1/(log(n+1)-log(n))). - _Thomas Ordowski_, Oct 10 2014",
"a(k) = det(S(2,k,1)). - _Ryan Stees_, Dec 15 2014",
"a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + ...). - _Pierre CAMI_, Jan 22 2015",
"a(n) = Sum_{m=0..n-1} Stirling1(n-1,m)*Bell(m+1), for n \u003e= 1. This corresponds to Bell(m+1) = Sum_{k=0..m} Stirling2(m, k)*(k+1), for m \u003e= 0, from the fact that Stirling2*Stirling1 = identity matrix. See A048993, A048994 and A000110. - _Wolfdieter Lang_, Feb 03 2015",
"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",
"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",
"a(n) = floor(1/(Pi/2-arctan(n))). - _Clark Kimberling_, Mar 11 2020",
"a(n) = Sum_{d|n} mu(n/d)*sigma(d). - _Ridouane Oudra_, Oct 03 2020",
"a(n) = Sum_{k=1..n} phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 09 2021",
"a(n) = S(n-1, 2), with the Chebyshev S-polynomials A049310. - _Wolfdieter Lang_, Mar 09 2023",
"From _Peter Bala_, Nov 02 2024: (Start)",
"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).",
"a(n) = Sum_{k = 0..3*n} (-1)^(n+k+1) * k * binomial(3*n+k, 2*k). (End)",
"T(n, k) = n + binomial(n+k-1, 2). - _Ruud H.G. van Tol_, Mar 10 2026"
],
"maple": [
"A000027 := n-\u003en; seq(A000027(n), n=1..100);"
],
"mathematica": [
"Range@ 77 (* _Robert G. Wilson v_, Mar 31 2015 *)"
],
"program": [
"(Magma) [ n : n in [1..100]];",
"(PARI) {a(n) = n};",
"(R) 1:100",
"(Shell) seq 1 100",
"(Haskell)",
"a000027 = id",
"a000027_list = [1..] -- _Reinhard Zumkeller_, May 07 2012",
"(Maxima) makelist(n, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */",
"(Python)",
"def A000027(n): return n # _Chai Wah Wu_, May 09 2022",
"(Julia) print([n for n in 1:280]) # _Paul Muljadi_, Apr 09 2024",
"(Perl) print join(\", \", 1..280) # _Paul Muljadi_, May 29 2024"
],
"xref": [
"A001477 = nonnegative numbers.",
"Partial sums of A000012.",
"Cf. A001478, A001906, A007931, A007932, A027641, A074909, A089353 (multisets), A178568, A194807.",
"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).",
"Cf. Boustrophedon transforms: A000737, A231179;",
"Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).",
"Cf. A048993, A048994, A000110 (see the Feb 03 2015 formula).",
"Cf. A289187, A000010, A008683, A000203, A049310."
],
"keyword": "core,nonn,easy,mult,tabl",
"offset": "1,2",
"author": "_N. J. A. Sloane_",
"ext": [
"Links edited by _Daniel Forgues_, Oct 07 2009"
],
"references": 2329,
"revision": 718,
"time": "2026-06-28T15:47:49-04:00",
"created": "1991-04-30T03:00:00-04:00"
}
]