cp's OEIS Frontend

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.

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A000027 The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

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 <= 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
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) < 1/n. - Clark Kimberling, Sep 28 2014
a(n) = least k such that Pi^2/6 - Sum_{h=1..k} 1/h^2 < 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>=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 >= 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

References

  • 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).

Links

Crossrefs

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.

Programs

Formula

a(2k+1) = A005408(k), k >= 0, a(2k) = A005843(k), k >= 1.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
Another g.f.: Sum_{n>0} 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>=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 <= i,j <= 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 > 1.
a(n) = Product_{k=1..n-1} (2*sin(Pi*k/(2*n)))^2, n > 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 -> 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>=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 >= 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
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)

Extensions

Links edited by Daniel Forgues, Oct 07 2009.

A001477 The nonnegative integers.

Original entry on oeis.org

0, 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
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Although this is a list, and lists normally have offset 1, it seems better to make an exception in this case. - N. J. A. Sloane, Mar 13 2010
The subsequence 0,1,2,3,4 gives the known values of n such that 2^(2^n)+1 is a prime (see A019434, the Fermat primes). - N. J. A. Sloane, Jun 16 2010
Also: The identity map, defined on the set of nonnegative integers. The restriction to the positive integers yields the sequence A000027. - M. F. Hasler, Nov 20 2013
The number of partitions of 2n into exactly 2 parts. - Colin Barker, Mar 22 2015
The number of orbits of Aut(Z^7) as function of the infinity norm n of the representative lattice point of the orbit, when the cardinality of the orbit is equal to 8960 or 168.- Philippe A.J.G. Chevalier, Dec 29 2015
Partial sums give A000217. - Omar E. Pol, Jul 26 2018
First differences are A000012 (the "all 1's" sequence). - M. F. Hasler, May 30 2020
See A061579 for the transposed infinite square matrix, or triangle with rows reversed. - M. F. Hasler, Nov 09 2021
This is the unique sequence (a(n)) that satisfies the inequality a(n+1) > a(a(n)) for all n in N. This simple and surprising result comes from the 6th problem proposed by Bulgaria during the second day of the 19th IMO (1977) in Belgrade (see link and reference). - Bernard Schott, Jan 25 2023

Examples

			Triangular view:
   0
   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

References

  • Maurice Protat, Des Olympiades à l'Agrégation, suite vérifiant f(n+1) > f(f(n)), Problème 7, pp. 31-32, Ellipses, Paris 1997.

Links

Crossrefs

Cf. A000027 (n>=1).
Cf. A000012 (first differences).
Partial sums of A057427. - Jeremy Gardiner, Sep 08 2002
Cf. A038608 (alternating signs), A001787 (binomial transform).
Cf. A055112.
Cf. Boustrophedon transforms: A231179, A000737.
Cf. A245422.
Number of orbits of Aut(Z^7) as function of the infinity norm A000579, A154286, A102860, A002412, A045943, A115067, A008586, A008585, A005843, A000217.
When written as an array, the rows/columns are A000217, A000124, A152948, A152950, A145018, A167499, A166136, A167487... and A000096, A034856, A055998, A046691, A052905, A055999... (with appropriate offsets); cf. analogous lists for A000027 in A185787.
Cf. A000290.
Cf. A061579 (transposed matrix / reversed triangle).

Programs

Formula

a(n) = n.
a(0) = 0, a(n) = a(n-1) + 1.
G.f.: x/(1-x)^2.
Multiplicative with a(p^e) = p^e. - David W. Wilson, Aug 01 2001
When seen as array: T(k, n) = n + (k+n)*(k+n+1)/2. Main diagonal is 2*n*(n+1) (A046092), antidiagonal sums are n*(n+1)*(n+2)/2 (A027480). - Ralf Stephan, Oct 17 2004
Dirichlet generating function: zeta(s-1). - Franklin T. Adams-Watters, Sep 11 2005
E.g.f.: x*e^x. - Franklin T. Adams-Watters, Sep 11 2005
a(0)=0, a(1)=1, a(n) = 2*a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
Alternating partial sums give A001057 = A000217 - 2*(A008794). - Eric Desbiaux, Oct 28 2008
a(n) = 2*A080425(n) + 3*A008611(n-3), n>1. - Eric Desbiaux, Nov 15 2009
a(n) = A007966(n)*A007967(n). - Reinhard Zumkeller, Jun 18 2011
a(n) = Sum_{k>=0} A030308(n,k)*2^k. - Philippe Deléham, Oct 20 2011
a(n) = 2*A028242(n-1) + (-1)^n*A000034(n-1). - R. J. Mathar, Jul 20 2012
a(n+1) = det(C(i+1,j), 1 <= i, j <= n), where C(n,k) are binomial coefficients. - Mircea Merca, Apr 06 2013
a(n-1) = floor(n/e^(1/n)) for n > 0. - Richard R. Forberg, Jun 22 2013
a(n) = A000027(n) for all n>0.
a(n) = floor(cot(1/(n+1))). - Clark Kimberling, Oct 08 2014
a(0)=0, a(n>0) = 2*z(-1)^[( |z|/z + 3 )/2] + ( |z|/z - 1 )/2 for z = A130472(n>0); a 1 to 1 correspondence between integers and naturals. - Adriano Caroli, Mar 29 2015
G.f. as triangle: x*(1 + (x^2 - 5*x + 2)*y + x*(2*x - 1)*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Jul 22 2025

A000737 Boustrophedon transform of natural numbers, cf. A000027.

Original entry on oeis.org

1, 3, 8, 21, 60, 197, 756, 3367, 17136, 98153, 624804, 4375283, 33424512, 276622829, 2465449252, 23543304919, 239810132288, 2595353815825, 29740563986500, 359735190398875, 4580290700420064, 61233976084442741
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A231179.

Programs

  • Haskell
    a000737 n = sum $ zipWith (*) (a109449_row n) [1..]
-- Reinhard Zumkeller, Nov 05 2013
  • Mathematica
    CoefficientList[Series[(1+x)*(Tan[x]+1/Cos[x])* E^x, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 02 2013 *)
t[n_, 0] := n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 30, 0] (* Jean-François Alcover, Feb 12 2016 *)
  • Python
    from itertools import count, accumulate, islice
def A000737_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000737_list = list(islice(A000737_gen(),40)) # Chai Wah Wu, Jun 12 2022
  • Sage
    # Algorithm of L. Seidel (1877)
def A000737_list(n) :
    R = []; A = {-1:0, 0:0}
    k = 0; e = 1
    for i in range(n) :
        Am = i+1
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        # To trace the algorithm remove the comment sign.
        # print([A[z] for z in (-i//2..i//2)])
        R.append(A[e*i//2])
    return R
A000737_list(10) # Peter Luschny, Jun 02 2012

Formula

E.g.f.: (1 + x)*(tan x + sec x)*exp(x).
a(n) ~ n! * (Pi + 2)*exp(Pi/2)*2^(n+1)/Pi^(n+1). - Vaclav Kotesovec, Oct 02 2013

A231200 Boustrophedon transform of even numbers.

Original entry on oeis.org

0, 2, 8, 24, 72, 240, 924, 4116, 20944, 119952, 763540, 5346748, 40845816, 338041704, 3012855356, 28770647220, 293055401888, 3171602665696, 36343889387172, 439607533130732, 5597256953340360, 74829813397495128, 1048039052970587788, 15345654816688856484
Offset: 0

Views

Author

Reinhard Zumkeller, Nov 05 2013

Keywords

Links

Crossrefs

Cf. A005843, A000754.

Programs

  • Haskell
    a231200 n = sum $ zipWith (*) (a109449_row n) $ [0, 2 ..]
  • Mathematica
    T[n_, k_] := SeriesCoefficient[(1+Sin[x])/Cos[x], {x, 0, n-k}] n!/k!;
a[n_] := 2 Sum[k T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Jun 28 2019 *)
  • Python
    from itertools import accumulate, count, islice
def A231200_gen(): # generator of terms
    blist = tuple()
    for i in count(0,2):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A231200_list = list(islice(A231200_gen(),40)) # Chai Wah Wu, Jun 12 2022

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*k*2.
a(n) = 2*A231179(n).
E.g.f.: 2*x*exp(x)*(sec(x) + tan(x)). - Ilya Gutkovskiy, Sep 27 2017

A000660 Boustrophedon transform of 1,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 5, 14, 41, 136, 523, 2330, 11857, 67912, 432291, 3027166, 23125673, 191389108, 1705788659, 16289080922, 165919213089, 1795666675824, 20576824369027, 248892651678198, 3168999664907705, 42366404751871660, 593368400878431795, 8688251294851280594
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Links

Crossrefs

Cf. A000111, A028310, A109449, A231179.

Programs

  • Haskell
    a000660 n = sum $ zipWith (*) (a109449_row n) (1 : [1..])
-- Reinhard Zumkeller, Nov 04 2013
  • Maple
    seq(coeff(series(factorial(n)*(x*exp(x)+1)*(sec(x)+tan(x)), x,n+1),x,n),n=0..25); # Muniru A Asiru, Jul 30 2018
  • Mathematica
    a[n_] := n! SeriesCoefficient[(1+x Exp[x])(1+Sin[x])/Cos[x], {x, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 30 2018, after Sergei N. Gladkovskii *)
  • Python
    from itertools import accumulate, count, islice
def A000660_gen(): # generator of terms
    yield 1
    blist = (1,)
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=i)))[-1]
A000660_list = list(islice(A000660_gen(),40)) # Chai Wah Wu, Jun 12 2022
  • Sage
    # Algorithm of L. Seidel (1877)
def A000660_list(n) :
    R = []; A = {-1:0, 0:1}
    k = 0; e = 1
    for i in range(n) :
        Am = i
        A[k + e] = 0
        e = -e
        for j in (0..i) :
            Am += A[k]
            A[k] = Am
            k += e
        print([A[z] for z in (-i//2..i//2)])
        R.append(A[e*i//2])
    return R
A000660_list(10) # Peter Luschny, Jun 02 2012

Formula

a(n) = Sum_{k=0..n} A109449(n,k)*A028310(k). - Reinhard Zumkeller, Nov 04 2013
E.g.f.: (x*exp(x) + 1)*(sec(x) + tan(x)). - Sergei N. Gladkovskii, Oct 28 2014
a(n) = A231179(n) + A000111(n). - Sergei N. Gladkovskii, Oct 28 2014
a(n) ~ n! * (2 + Pi*exp(Pi/2)) * (2/Pi)^(n+1). - Vaclav Kotesovec, Jun 12 2015

A294033 Triangle read by rows, expansion of exp(x*z)*z*(tanh(z) + sech(z)), T(n, k) for n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 2, 2, -3, 6, 3, -8, -12, 12, 4, 25, -40, -30, 20, 5, 96, 150, -120, -60, 30, 6, -427, 672, 525, -280, -105, 42, 7, -2176, -3416, 2688, 1400, -560, -168, 56, 8, 12465, -19584, -15372, 8064, 3150, -1008, -252, 72, 9, 79360, 124650, -97920, -51240, 20160, 6300, -1680, -360, 90, 10, -555731, 872960, 685575, -359040, -140910, 44352, 11550, -2640, -495, 110, 11
Offset: 1

Views

Author

Peter Luschny, Oct 24 2017

Keywords

Examples

			Triangle starts:
  [1][   1]
  [2][   2,   2]
  [3][  -3,   6,    3]
  [4][  -8, -12,   12,    4]
  [5][  25, -40,  -30,   20,    5]
  [6][  96, 150, -120,  -60,   30,  6]
  [7][-427, 672,  525, -280, -105, 42, 7]

Crossrefs

T(n, 0) = signed A065619. Row sums of abs(T(n,k)) = A231179.
Diagonals A000027, A002378, A027480, A162668.
A003506 (m=1), this seq. (m=2), A294034 (m=3).
Cf. A000111, A247453.

Programs

  • Maple
    gf := exp(x*z)*z*(tanh(z)+sech(z)):
s := n -> n!*coeff(series(gf,z,n+2),z,n):
C := n -> PolynomialTools:-CoefficientList(s(n),x):
ListTools:-FlattenOnce([seq(C(n), n=1..7)]);
# Alternatively:
T := (n, k) -> `if`(n = k+1, n,
(k+1)*binomial(n,k+1)*2^(n-k-1)*(euler(n-k-1, 1/2)+euler(n-k-1, 1))):
for n from 1 to 7 do seq(T(n,k), k=0..n-1) od;
  • Mathematica
    L[0] := 1; L[n_] := (-1)^Binomial[n, 2] 2 Abs[PolyLog[-n, -I]];
p[n_] := n Sum[Binomial[n - 1, k - 1] L[k - 1] x^(n - k), {k, 0, n}];
Table[CoefficientList[p[n], x], {n, 1, 11}] // Flatten

Formula

T(n, k) = (k+1)*binomial(n,k+1)*2^(n-k-1)*(Euler(n-k-1, 1/2) + Euler(n-k-1, 1)) for 0 <= k <= n-2.
T(n, k) is the coefficient of x^k of the polynomial p(n) = n*Sum_{k=1..n} binomial(n-1, k-1)*L(k-1)*x^(n-k) and L(n) = (-1)^binomial(n,2)*A000111(n). In particular n divides T(n, k).

A337443 E.g.f.: (1 + x) * exp(x) / (sec(x) + tan(x)).

Original entry on oeis.org

1, 1, 0, -1, -4, -5, -20, 9, -208, 855, -6180, 41549, -321792, 2651155, -23664420, 225865769, -2301032256, 24901626095, -285356879940, 3451591584869, -43947119045600, 587529302036875, -8228722825167940, 120487046847246049, -1840906518665985824
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 27 2020

Keywords

Comments

Inverse boustrophedon transform of positive natural numbers.

Links

Crossrefs

Cf. A000027, A000111, A000737, A034428, A231179.

Programs

  • Mathematica
    nmax = 24; CoefficientList[Series[(1 + x) Exp[x]/(Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] - t[n - 1, n - k]; a[n_] := t[n, n]; Table[a[n], {n, 0, 24}]
  • Python
    from itertools import count, islice, accumulate
from operator import sub
def A337443_gen(): # generator of terms
    blist = tuple()
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=i)))[-1]
A337443_list = list(islice(A337443_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (k+1) * A000111(n-k).
Showing 1-7 of 7 results.

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