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%I M0472 N0173 #650 Feb 11 2024 15:06:56
%S 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 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,
%U 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 The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.
%C 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 a(n) is smallest positive integer which is consistent with sequence being monotonically increasing and satisfying a(a(n)) = n (cf. A007378).
%C Inverse Euler transform of A000219.
%C 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 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 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 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 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 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 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 "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 Binomial transform of A019590, inverse binomial transform of A001792. - _Philippe Deléham_, Oct 24 2007
%C 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 a(n) is also the mean of the first n odd integers. - _Ian Kent_, Dec 23 2008
%C Equals INVERTi transform of A001906, the even-indexed Fibonacci numbers starting (1, 3, 8, 21, 55, ...). - _Gary W. Adamson_, Jun 05 2009
%C These are also the 2-rough numbers: positive integers that have no prime factors less than 2. - _Michael B. Porter_, Oct 08 2009
%C 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 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 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 a(n) is also the dimension of the irreducible representations of the Lie algebra sl(2). - _Leonid Bedratyuk_, Jan 04 2010
%C Floyd's triangle read by rows. - _Paul Muljadi_, Jan 25 2010
%C Number of numbers between k and 2k where k is an integer. - _Giovanni Teofilatto_, Mar 26 2010
%C 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 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 Number of n-digit numbers the binary expansion of which contains one run of 1's. - _Vladimir Shevelev_, Jul 30 2010
%C From _Clark Kimberling_, Jan 29 2011: (Start)
%C Let T denote the "natural number array A000027":
%C 1 2 4 7 ...
%C 3 5 8 12 ...
%C 6 9 13 18 ...
%C 10 14 19 25 ...
%C 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 The Stern polynomial B(n,x) evaluated at x=2. See A125184. - _T. D. Noe_, Feb 28 2011
%C 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 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 Number of partitions of 2n+1 into exactly two parts. - _Wesley Ivan Hurt_, Jul 15 2013
%C 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 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 Engel expansion of e-1 (A091131 = 1.71828...). - _Jaroslav Krizek_, Jan 23 2014
%C 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 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 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 a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 1/n. - _Clark Kimberling_, Oct 02 2014
%C 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 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 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 a(n) is the number of compositions of n+2 into n parts avoiding the part 2. - _Milan Janjic_, Jan 07 2016
%C Does not satisfy Benford's law [Berger-Hill, 2017] - _N. J. A. Sloane_, Feb 07 2017
%C 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 The arithmetic function v_1(n,1) as defined in A289197. - _Robert Price_, Aug 22 2017
%C 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 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 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 (1, 2, 3, 4, 5, ...) is the fourth INVERT transform of (1, -2, 3, -4, 5, ...). - _Gary W. Adamson_, Jul 15 2019
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 1.
%D T. M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, 1990, page 25.
%D W. Fulton and J. Harris, Representation theory: a first course, (1991), page 149. [From _Leonid Bedratyuk_, Jan 04 2010]
%D I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
%D R. E. Schwartz, You Can Count on Monsters: The First 100 numbers and Their Characters, A. K. Peters and MAA, 2010.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A000027/b000027.txt">Table of n, a(n) for n = 1..500000</a> [a large file]
%H Archimedes Laboratory, <a href="http://www.archimedes-lab.org/numbers/Num1_200.html">What's special about this number?</a>
%H 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 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 James Barton, <a href="http://www.virtuescience.com/number-list.html">The Numbers</a>
%H 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 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 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 C. K. Caldwell, <a href="https://t5k.org/curios">Prime Curios</a>
%H Case and Abiessu, <a href="http://everything2.net/index.pl?node_id=17633&displaytype=printable&lastnode_id=17633">interesting number</a>
%H S. Crandall, <a href="http://tingilinde.typepad.com/starstuff/2005/11/significant_int.html">notes on interesting digital ephemera</a>
%H O. Curtis, <a href="http://users.pipeline.com.au/owen/Numbers.html">Interesting Numbers</a>
%H 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 Walter Felscher, <a href="http://sunsite.utk.edu/math_archives/.http/hypermail/historia/may99/0210.html">Historia Matematica Mailing List Archive.</a>
%H P. Flajolet and R. Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/books.html">Analytic Combinatorics</a>, 2009; see page 371.
%H 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 E. Friedman, <a href="https://erich-friedman.github.io/numbers.html">What's Special About This Number?</a>
%H R. K. Guy, <a href="/A000346/a000346.pdf">Letter to N. J. A. Sloane</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>
%H 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 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 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 R. Munafo, <a href="http://www.mrob.com/pub/math/numbers.html">Notable Properties of Specific Numbers</a>
%H 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 R. Phillips, <a href="http://richardphillips.org.uk/number/Num1.htm">Numbers from one to thirty-one</a>
%H 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 G. Villemin's Almanac of Numbers, <a href="http://villemin.gerard.free.fr/aNombre/Nb0a50.htm">NOMBRES en BREF (in French)</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NaturalNumber.html">Natural Number</a>, <a href="http://mathworld.wolfram.com/PositiveInteger.html">Positive Integer</a>, <a href="http://mathworld.wolfram.com/CountingNumber.html">Counting Number</a> <a href="http://mathworld.wolfram.com/Composition.html">Composition</a>, <a href="http://mathworld.wolfram.com/Davenport-SchinzelSequence.html">Davenport-Schinzel Sequence</a>, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>, <a href="http://mathworld.wolfram.com/N.html">N</a>, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>, <a href="http://mathworld.wolfram.com/WholeNumber.html">Whole Number</a>, <a href="http://mathworld.wolfram.com/EngelExpansion.html">Engel Expansion</a>, and <a href="http://mathworld.wolfram.com/TrinomialCoefficient.html">Trinomial Coefficient</a>
%H 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 Robert G. Wilson v, <a href="/A000027/a000027.txt">English names for the numbers from 0 to 11159 without spaces or hyphens</a>
%H 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 <a href="/index/Cor#core">Index entries for "core" sequences</a>
%H <a href="/index/Aa#aan">Index entries for sequences of the a(a(n)) = 2n family</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Be#Benford">Index entries for sequences related to Benford's law</a>
%F a(2k+1) = A005408(k), k >= 0, a(2k) = A005843(k), k >= 1.
%F Multiplicative with a(p^e) = p^e. - _David W. Wilson_, Aug 01 2001
%F Another g.f.: Sum_{n>0} phi(n)*x^n/(1-x^n) (Apostol).
%F 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 Dirichlet generating function: zeta(s-1). - _Franklin T. Adams-Watters_, Sep 11 2005
%F G.f.: x/(1-x)^2. E.g.f.: x*exp(x). a(n)=n. a(-n)=-a(n).
%F 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 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 Convolution of A000012 (the all-ones sequence) with itself. - _Tanya Khovanova_, Jun 22 2007
%F 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 a(n) = A000720(A000040(n)). - _Juri-Stepan Gerasimov_, Nov 29 2009
%F a(n+1) = Sum_{k=0..n} A101950(n,k). - _Philippe Deléham_, Feb 10 2012
%F a(n) = Sum_{d | n} phi(d) = Sum_{d | n} A000010(d). - _Jaroslav Krizek_, Apr 20 2012
%F 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 a(n) = det(binomial(i+1,j), 1 <= i,j <= n). - _Mircea Merca_, Apr 06 2013
%F 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 From _Wolfdieter Lang_, Oct 09 2013: (Start)
%F a(n) = Product_{k=1..n-1} 2*sin(Pi*k/n), n > 1.
%F a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 1.
%F 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 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 a(n) = A052410(n)^A052409(n). - _Reinhard Zumkeller_, Apr 06 2014
%F a(n) = Sum_{k=1..n^2+2*n} 1/(sqrt(k)+sqrt(k+1)). - _Pierre CAMI_, Apr 25 2014
%F a(n) = floor(1/sin(1/n)) = floor(cot(1/(n+1))) = ceiling(cot(1/n)). - _Clark Kimberling_, Oct 08 2014
%F a(n) = floor(1/(log(n+1)-log(n))). - _Thomas Ordowski_, Oct 10 2014
%F a(k) = det(S(2,k,1)). - _Ryan Stees_, Dec 15 2014
%F a(n) = 1/(1/(n+1) + 1/(n+1)^2 + 1/(n+1)^3 + ...). - _Pierre CAMI_, Jan 22 2015
%F 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 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 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 a(n) = floor(1/(Pi/2-arctan(n))). - _Clark Kimberling_, Mar 11 2020
%F a(n) = Sum_{d|n} mu(n/d)*sigma(d). - _Ridouane Oudra_, Oct 03 2020
%F a(n) = Sum_{k=1..n} phi(gcd(n,k))/phi(n/gcd(n,k)). - _Richard L. Ollerton_, May 09 2021
%F a(n) = S(n-1, 2), with the Chebyshev S-polynomials A049310. - _Wolfdieter Lang_, Mar 09 2023
%p A000027 := n->n; seq(A000027(n), n=1..100);
%t Range@ 77 (* _Robert G. Wilson v_, Mar 31 2015 *)
%o (Magma) [ n : n in [1..100]];
%o (PARI) {a(n) = n};
%o (R) 1:100
%o (Shell) seq 1 100
%o (Haskell)
%o a000027 = id
%o a000027_list = [1..] -- _Reinhard Zumkeller_, May 07 2012
%o (Maxima) makelist(n, n, 1, 30); /* _Martin Ettl_, Nov 07 2012 */
%o (Python)
%o def A000027(n): return n # _Chai Wah Wu_, May 09 2022
%Y A001477 = nonnegative numbers.
%Y Partial sums of A000012.
%Y Cf. A001478, A001906, A007931, A007932, A027641, A074909, A089353 (multisets), A178568, A194807.
%Y 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 Cf. Boustrophedon transforms: A000737, A231179;
%Y Cf. A038722 (mirrored when seen as triangle), A056011 (boustrophedon).
%Y Cf. A048993, A048994, A000110 (see the Feb 03 2015 formula).
%Y Cf. A289187.
%Y Cf. A000010, A008683, A000203, A049310.
%K core,nonn,easy,mult,tabl
%O 1,2
%A _N. J. A. Sloane_
%E Links edited by _Daniel Forgues_, Oct 07 2009
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