A007395 -id:A007395 - cp's OEIS frontend

A000012 The simplest sequence of positive numbers: the all 1's sequence.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, May 16 1994

Number of ways of writing n as a product of primes.
Number of ways of writing n as a sum of distinct powers of 2.
Continued fraction for golden ratio A001622.
Partial sums of A000007 (characteristic function of 0). - Jeremy Gardiner, Sep 08 2002
An example of an infinite sequence of positive integers whose distinct pairwise concatenations are all primes! - Don Reble, Apr 17 2005
Binomial transform of A000007; inverse binomial transform of A000079. - Philippe Deléham, Jul 07 2005
A063524(a(n)) = 1. - Reinhard Zumkeller, Oct 11 2008
For n >= 0, let M(n) be the matrix with first row = (n n+1) and 2nd row = (n+1 n+2). Then a(n) = absolute value of det(M(n)). - K.V.Iyer, Apr 11 2009
The partial sums give the natural numbers (A000027). - Daniel Forgues, May 08 2009
From Enrique Pérez Herrero, Sep 04 2009: (Start)
a(n) is also tau_1(n) where tau_2(n) is A000005.
a(n) is a completely multiplicative arithmetical function.
a(n) is both squarefree and a perfect square. See A005117 and A000290. (End)
Also smallest divisor of n. - Juri-Stepan Gerasimov, Sep 07 2009
Also decimal expansion of 1/9. - Enrique Pérez Herrero, Sep 18 2009; corrected by Klaus Brockhaus, Apr 02 2010
a(n) is also the number of complete graphs on n nodes. - Pablo Chavez (pchavez(AT)cmu.edu), Sep 15 2009
Totally multiplicative sequence with a(p) = 1 for prime p. Totally multiplicative sequence with a(p) = a(p-1) for prime p. - Jaroslav Krizek, Oct 18 2009
n-th prime minus phi(prime(n)); number of divisors of n-th prime minus number of perfect partitions of n-th prime; the number of perfect partitions of n-th prime number; the number of perfect partitions of n-th noncomposite number. - Juri-Stepan Gerasimov, Oct 26 2009
For all n>0, the sequence of limit values for a(n) = n!*Sum_{k>=n} k/(k+1)!. Also, a(n) = n^0. - Harlan J. Brothers, Nov 01 2009
a(n) is also the number of 0-regular graphs on n vertices. - Jason Kimberley, Nov 07 2009
Differences between consecutive n. - Juri-Stepan Gerasimov, Dec 05 2009
From Matthew Vandermast, Oct 31 2010: (Start)
1) When sequence is read as a regular triangular array, T(n,k) is the coefficient of the k-th power in the expansion of (x^(n+1)-1)/(x-1).
2) Sequence can also be read as a uninomial array with rows of length 1, analogous to arrays of binomial, trinomial, etc., coefficients. In a q-nomial array, T(n,k) is the coefficient of the k-th power in the expansion of ((x^q -1)/(x-1))^n, and row n has a sum of q^n and a length of (q-1)*n + 1. (End)
The number of maximal self-avoiding walks from the NW to SW corners of a 2 X n grid.
When considered as a rectangular array, A000012 is a member of the chain of accumulation arrays that includes the multiplication table A003991 of the positive integers. The chain is ... < A185906 < A000007 < A000012 < A003991 < A098358 < A185904 < A185905 < ... (See A144112 for the definition of accumulation array.) - Clark Kimberling, Feb 06 2011
a(n) = A007310(n+1) (Modd 3) := A193680(A007310(n+1)), n>=0. For general Modd n (not to be confused with mod n) see a comment on A203571. The nonnegative members of the three residue classes Modd 3, called [0], [1], and [2], are shown in the array A088520, if there the third row is taken as class [0] after inclusion of 0. - Wolfdieter Lang, Feb 09 2012
Let M = Pascal's triangle without 1's (A014410) and V = a variant of the Bernoulli numbers A027641 but starting [1/2, 1/6, 0, -1/30, ...]. Then M*V = [1, 1, 1, 1, ...]. - Gary W. Adamson, Mar 05 2012
As a lower triangular array, T is an example of the fundamental generalized factorial matrices of A133314. Multiplying each n-th diagonal by t^n gives M(t) = I/(I-t*S) = I + t*S + (t*S)^2 + ... where S is the shift operator A129184, and T = M(1). The inverse of M(t) is obtained by multiplying the first subdiagonal of T by -t and the other subdiagonals by zero, so A167374 is the inverse of T. Multiplying by t^n/n! gives exp(t*S) with inverse exp(-t*S). - Tom Copeland, Nov 10 2012
The original definition of the meter was one ten-millionth of the distance from the Earth's equator to the North Pole. According to that historical definition, the length of one degree of latitude, that is, 60 nautical miles, would be exactly 111111.111... meters. - Jean-François Alcover, Jun 02 2013
Deficiency of 2^n. - Omar E. Pol, Jan 30 2014
Consider n >= 1 nonintersecting spheres each with surface area S. Define point p on sphere S_i to be a "public point" if and only if there exists a point q on sphere S_j, j != i, such that line segment pq INTERSECT S_i = {p} and pq INTERSECT S_j = {q}; otherwise, p is a "private point". The total surface area composed of exactly all private points on all n spheres is a(n)*S = S. ("The Private Planets Problem" in Zeitz.) - Rick L. Shepherd, May 29 2014
For n>0, digital roots of centered 9-gonal numbers (A060544). - Colin Barker, Jan 30 2015
Product of nonzero digits in base-2 representation of n. - Franklin T. Adams-Watters, May 16 2016
Alternating row sums of triangle A104684. - Wolfdieter Lang, Sep 11 2016
A fixed point of the run length transform. - Chai Wah Wu, Oct 21 2016
Length of period of continued fraction for sqrt(A002522) or sqrt(A002496). - A.H.M. Smeets, Oct 10 2017
a(n) is also the determinant of the (n+1) X (n+1) matrix M defined by M(i,j) = binomial(i,j) for 0 <= i,j <= n, since M is a lower triangular matrix with main diagonal all 1's. - Jianing Song, Jul 17 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = min(i,j) for 1 <= i,j <= n (see Xavier Merlin reference). - Bernard Schott, Dec 05 2018
a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = tau(gcd(i,j)) for 1 <= i,j <= n (see De Koninck & Mercier reference). - Bernard Schott, Dec 08 2020

			1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...)))) = A001622.
1/9 = 0.11111111111111...
From _Wolfdieter Lang_, Feb 09 2012: (Start)
Modd 7 for nonnegative odd numbers not divisible by 3:
A007310: 1, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, ...
Modd 3:  1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
(End)

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 186.
J.-M. De Koninck & A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 692 pp. 90 and 297, Ellipses, Paris, 2004.
Xavier Merlin, Méthodix Algèbre, Exercice 1-a), page 153, Ellipses, Paris, 1995.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 277, 284.
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 55.
Paul Zeitz, The Art and Craft of Mathematical Problem Solving, The Great Courses, The Teaching Company, 2010 (DVDs and Course Guidebook, Lecture 6: "Pictures, Recasting, and Points of View", pp. 32-34).

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 [Useful when plotting one sequence against another.]
Jeremiah Bartz, Bruce Dearden, and Joel Iiams, Classes of Gap Balancing Numbers, arXiv:1810.07895 [math.NT], 2018.
Harlan Brothers, Factorial: Summation (formula 06.01.23.0002), The Wolfram Functions Site - _Harlan J. Brothers_, Nov 01 2009
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 172. Book's website
L. B. W. Jolley, Summation of Series, Dover, 1961
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Robert Price, Comments on A000012 concerning Elementary Cellular Automata, Jan 31 2016
N. J. A. Sloane, Illustration of initial terms
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
Eric Weisstein's World of Mathematics, Golden Ratio
Eric Weisstein's World of Mathematics, Chromatic Number
Eric Weisstein's World of Mathematics, Graph Cycle
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
G. Xiao, Contfrac
Index entries for "core" sequences
Index entries for characteristic functions
Index entries for continued fractions for constants
Index to divisibility sequences
Index entries for related partition-counting sequences
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for sequences that are fixed points of mappings

Cf. A000004, A007395, A010701, A000027, A027641, A014410, A211216, A212393, A060544, A051801, A104684.
For other q-nomial arrays, see A007318, A027907, A008287, A035343, A063260, A063265, A171890. - Matthew Vandermast, Oct 31 2010
Cf. A097805, A118800, A130595, A167374, A008284 (multisets).

a000012 = const 1
a000012_list = repeat 1 -- Reinhard Zumkeller, May 07 2012

Magma
```
[1 : n in [0..100]];
```
Maple
```
seq(1, i=0..150);
```

Array[1 &, 50] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Dec 26 2006 *)

makelist(1, n, 1, 30); /* Martin Ettl, Nov 07 2012 */

PARI
```
{a(n) = 1};
```

print([1 for n in range(90)]) # Michael S. Branicky, Apr 04 2022

a(n) = 1.
G.f.: 1/(1-x).
E.g.f.: exp(x).
G.f.: Product_{k>=0} (1 + x^(2^k)). - Zak Seidov, Apr 06 2007
Completely multiplicative with a(p^e) = 1.
Regarded as a square array by antidiagonals, g.f. 1/((1-x)(1-y)), e.g.f. Sum T(n,m) x^n/n! y^m/m! = e^{x+y}, e.g.f. Sum T(n,m) x^n y^m/m! = e^y/(1-x). Regarded as a triangular array, g.f. 1/((1-x)(1-xy)), e.g.f. Sum T(n,m) x^n y^m/m! = e^{xy}/(1-x). - Franklin T. Adams-Watters, Feb 06 2006
Dirichlet g.f.: zeta(s). - Ilya Gutkovskiy, Aug 31 2016
a(n) = Sum_{l=1..n} (-1)^(l+1)*2*cos(Pi*l/(2*n+1)) = 1 identically in n >= 1 (for n=0 one has 0 from the undefined sum). From the Jolley reference, (429) p. 80. Interpretation: consider the n segments between x=0 and the n positive zeros of the Chebyshev polynomials S(2*n, x) (see A049310). Then the sum of the lengths of every other segment starting with the one ending in the largest zero (going from the right to the left) is 1. - Wolfdieter Lang, Sep 01 2016
As a lower triangular matrix, T = M*T^(-1)*M = M*A167374*M, where M(n,k) = (-1)^n A130595(n,k). Note that M = M^(-1). Cf. A118800 and A097805. - Tom Copeland, Nov 15 2016

A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

Original entry on oeis.org

0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550

Offset: 0

N. J. A. Sloane

4*a(n) + 1 are the odd squares A016754(n).
The word "pronic" (used by Dickson) is incorrect. - Michael Somos
According to the 2nd edition of Webster, the correct word is "promic". - R. K. Guy
a(n) is the number of minimal vectors in the root lattice A_n (see Conway and Sloane, p. 109).
Let M_n denote the n X n matrix M_n(i, j) = (i + j); then the characteristic polynomial of M_n is x^(n-2) * (x^2 - a(n)*x - A002415(n)). - Benoit Cloitre, Nov 09 2002
The greatest LCM of all pairs (j, k) for j < k <= n for n > 1. - Robert G. Wilson v, Jun 19 2004
First differences are a(n+1) - a(n) = 2*n + 2 = 2, 4, 6, ... (while first differences of the squares are (n+1)^2 - n^2 = 2*n + 1 = 1, 3, 5, ...). - Alexandre Wajnberg, Dec 29 2005
25 appended to these numbers corresponds to squares of numbers ending in 5 (i.e., to squares of A017329). - Lekraj Beedassy, Mar 24 2006
A rapid (mental) multiplication/factorization technique -- a generalization of Lekraj Beedassy's comment: For all bases b >= 2 and positive integers n, c, d, k with c + d = b^k, we have (n*b^k + c)*(n*b^k + d) = a(n)*b^(2*k) + c*d. Thus the last 2*k base-b digits of the product are exactly those of c*d -- including leading 0(s) as necessary -- with the preceding base-b digit(s) the same as a(n)'s. Examples: In decimal, 113*117 = 13221 (as n = 11, b = 10 = 3 + 7, k = 1, 3*7 = 21, and a(11) = 132); in octal, 61*67 = 5207 (52 is a(6) in octal). In particular, for even b = 2*m (m > 0) and c = d = m, such a product is a square of this type. Decimal factoring: 5609 is immediately seen to be 71*79. Likewise, 120099 = 301*399 (k = 2 here) and 99990000001996 = 9999002*9999998 (k = 3). - Rick L. Shepherd, Jul 24 2021
Number of circular binary words of length n + 1 having exactly one occurrence of 01. Example: a(2) = 6 because we have 001, 010, 011, 100, 101 and 110. Column 1 of A119462. - Emeric Deutsch, May 21 2006
The sequence of iterated square roots sqrt(N + sqrt(N + ...)) has for N = 1, 2, ... the limit (1 + sqrt(1 + 4*N))/2. For N = a(n) this limit is n + 1, n = 1, 2, .... For all other numbers N, N >= 1, this limit is not a natural number. Examples: n = 1, a(1) = 2: sqrt(2 + sqrt(2 + ...)) = 1 + 1 = 2; n = 2, a(2) = 6: sqrt(6 + sqrt(6 + ...)) = 1 + 2 = 3. - Wolfdieter Lang, May 05 2006
Nonsquare integers m divisible by ceiling(sqrt(m)), except for m = 0. - Max Alekseyev, Nov 27 2006
The number of off-diagonal elements of an (n + 1) X (n + 1) matrix. - Artur Jasinski, Jan 11 2007
a(n) is equal to the number of functions f:{1, 2} -> {1, 2, ..., n + 1} such that for a fixed x in {1, 2} and a fixed y in {1, 2, ..., n + 1} we have f(x) <> y. - Aleksandar M. Janjic and Milan Janjic, Mar 13 2007
Numbers m >= 0 such that round(sqrt(m+1)) - round(sqrt(m)) = 1. - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that ceiling(2*sqrt(m+1)) - 1 = 1 + floor(2*sqrt(m)). - Hieronymus Fischer, Aug 06 2007
Numbers m >= 0 such that fract(sqrt(m+1)) > 1/2 and fract(sqrt(m)) < 1/2 where fract(x) is the fractional part (fract(x) = x - floor(x), x >= 0). - Hieronymus Fischer, Aug 06 2007
X values of solutions to the equation 4*X^3 + X^2 = Y^2. To find Y values: b(n) = n(n+1)(2n+1). - Mohamed Bouhamida, Nov 06 2007
Nonvanishing diagonal of A132792, the infinitesimal Lah matrix, so "generalized factorials" composed of a(n) are given by the elements of the Lah matrix, unsigned A111596, e.g., a(1)*a(2)*a(3) / 3! = -A111596(4,1) = 24. - Tom Copeland, Nov 20 2007
If Y is a 2-subset of an n-set X then, for n >= 2, a(n-2) is the number of 2-subsets and 3-subsets of X having exactly one element in common with Y. - Milan Janjic, Dec 28 2007
a(n) coincides with the vertex of a parabola of even width in the Redheffer matrix, directed toward zero. An integer p is prime if and only if for all integer k, the parabola y = kx - x^2 has no integer solution with 1 < x < k when y = p; a(n) corresponds to odd k. - Reikku Kulon, Nov 30 2008
The third differences of certain values of the hypergeometric function 3F2 lead to the squares of the oblong numbers i.e., 3F2([1, n + 1, n + 1], [n + 2, n + 2], z = 1) - 3*3F2([1, n + 2, n + 2], [n + 3, n + 3], z = 1) + 3*3F2([1, n + 3, n + 3], [n + 4, n + 4], z = 1) - 3F2([1, n + 4, n + 4], [n + 5, n + 5], z = 1) = (1/((n+2)*(n+3)))^2 for n = -1, 0, 1, 2, ... . See also A162990. - Johannes W. Meijer, Jul 21 2009
Generalized factorials, [a.(n!)] = a(n)*a(n-1)*...*a(0) = A010790(n), with a(0) = 1 are related to A001263. - Tom Copeland, Sep 21 2011
For n > 1, a(n) is the number of functions f:{1, 2} -> {1, ..., n + 2} where f(1) > 1 and f(2) > 2. Note that there are n + 1 possible values for f(1) and n possible values for f(2). For example, a(3) = 12 since there are 12 functions f from {1, 2} to {1, 2, 3, 4, 5} with f(1) > 1 and f(2) > 2. - Dennis P. Walsh, Dec 24 2011
a(n) gives the number of (n + 1) X (n + 1) symmetric (0, 1)-matrices containing two ones (see [Cameron]). - L. Edson Jeffery, Feb 18 2012
a(n) is the number of positions of a domino in a rectangled triangular board with both legs equal to n + 1. - César Eliud Lozada, Sep 26 2012
a(n) is the number of ordered pairs (x, y) in [n+2] X [n+2] with |x-y| > 1. - Dennis P. Walsh, Nov 27 2012
a(n) is the number of injective functions from {1, 2} into {1, 2, ..., n + 1}. - Dennis P. Walsh, Nov 27 2012
a(n) is the sum of the positive differences of the partition parts of 2n + 2 into exactly two parts (see example). - Wesley Ivan Hurt, Jun 02 2013
a(n)/a(n-1) is asymptotic to e^(2/n). - Richard R. Forberg, Jun 22 2013
Number of positive roots in the root system of type D_{n + 1} (for n > 2). - Tom Edgar, Nov 05 2013
Number of roots in the root system of type A_n (for n > 0). - Tom Edgar, Nov 05 2013
From Felix P. Muga II, Mar 18 2014: (Start)
a(m), for m >= 1, are the only positive integer values t for which the Binet-de Moivre formula for the recurrence b(n) = b(n-1) + t*b(n-2) with b(0) = 0 and b(1) = 1 has a root of a square. PROOF (as suggested by Wolfdieter Lang, Mar 26 2014): The sqrt(1 + 4t) appearing in the zeros r1 and r2 of the characteristic equation is (a positive) integer for positive integer t precisely if 4t + 1 = (2m + 1)^2, that is t = a(m), m >= 1. Thus, the characteristic roots are integers: r1 = m + 1 and r2 = -m.
Let m > 1 be an integer. If b(n) = b(n-1) + a(m)*b(n-2), n >= 2, b(0) = 0, b(1) = 1, then lim_{n->oo} b(n+1)/b(n) = m + 1. (End)
Cf. A130534 for relations to colored forests, disposition of flags on flagpoles, and colorings of the vertices (chromatic polynomial) of the complete graphs (here simply K_2). - Tom Copeland, Apr 05 2014
The set of integers k for which k + sqrt(k + sqrt(k + sqrt(k + sqrt(k + ...) ... is an integer. - Leslie Koller, Apr 11 2014
a(n-1) is the largest number k such that (n*k)/(n+k) is an integer. - Derek Orr, May 22 2014
Number of ways to place a domino and a singleton on a strip of length n - 2. - Ralf Stephan, Jun 09 2014
With offset 1, this appears to give the maximal number of crossings between n nonconcentric circles of equal radius. - Felix Fröhlich, Jul 14 2014
For n > 1, the harmonic mean of the n values a(1) to a(n) is n + 1. The lowest infinite sequence of increasing positive integers whose cumulative harmonic mean is integral. - Ian Duff, Feb 01 2015
a(n) is the maximum number of queens of one color that can coexist without attacking one queen of the opponent's color on an (n+2) X (n+2) chessboard. The lone queen can be placed in any position on the perimeter of the board. - Bob Selcoe, Feb 07 2015
With a(0) = 1, a(n-1) is the smallest positive number not in the sequence such that Sum_{i = 1..n} 1/a(i-1) has a denominator equal to n. - Derek Orr, Jun 17 2015
The positive members of this sequence are a proper subsequence of the so-called 1-happy couple products A007969. See the W. Lang link there, eq. (4), with Y_0 = 1, with a table at the end. - Wolfdieter Lang, Sep 19 2015
For n > 0, a(n) is the reciprocal of the area bounded above by y = x^(n-1) and below by y = x^n for x in the interval [0, 1]. Summing all such areas visually demonstrates the formula below giving Sum_{n >= 1} 1/a(n) = 1. - Rick L. Shepherd, Oct 26 2015
It appears that, except for a(0) = 0, this is the set of positive integers n such that x*floor(x) = n has no solution. (For example, to get 3, take x = -3/2.) - Melvin Peralta, Apr 14 2016
If two independent real random variables, x and y, are distributed according to the same exponential distribution: pdf(x) = lambda * exp(-lambda * x), lambda > 0, then the probability that n - 1 <= x/y < n is given by 1/a(n). - Andres Cicuttin, Dec 03 2016
a(n) is equal to the sum of all possible differences between n different pairs of consecutive odd numbers (see example). - Miquel Cerda, Dec 04 2016
a(n+1) is the dimension of the space of vector fields in the plane with polynomial coefficients up to order n. - Martin Licht, Dec 04 2016
It appears that a(n) + 3 is the area of the largest possible pond in a square (A268311). - Craig Knecht, May 04 2017
Also the number of 3-cycles in the (n+3)-triangular honeycomb acute knight graph. - Eric W. Weisstein, Jul 27 2017
Also the Wiener index of the (n+2)-wheel graph. - Eric W. Weisstein, Sep 08 2017
The left edge of a Floyd's triangle that consists of even numbers: 0; 2, 4; 6, 8, 10; 12, 14, 16, 18; 20, 22, 24, 26, 28; ... giving 0, 2, 6, 12, 20, ... The right edge generates A028552. - Waldemar Puszkarz, Feb 02 2018
a(n+1) is the order of rowmotion on a poset obtained by adjoining a unique minimal (or maximal) element to a disjoint union of at least two chains of n elements. - Nick Mayers, Jun 01 2018
From Juhani Heino, Feb 05 2019: (Start)
For n > 0, 1/a(n) = n/(n+1) - (n-1)/n.
For example, 1/6 = 2/3 - 1/2; 1/12 = 3/4 - 2/3.
Corollary of this:
Take 1/2 pill.
Next day, take 1/6 pill. 1/2 + 1/6 = 2/3, so your daily average is 1/3.
Next day, take 1/12 pill. 2/3 + 1/12 = 3/4, so your daily average is 1/4.
And so on. (End)
From Bernard Schott, May 22 2020: (Start)
For an oblong number m >= 6 there exists a Euclidean division m = d*q + r with q < r < d which are in geometric progression, in this order, with a common integer ratio b. For b >= 2 and q >= 1, the Euclidean division is m = qb*(qb+1) = qb^2 * q + qb where (q, qb, qb^2) are in geometric progression.
Some examples with distinct ratios and quotients:
   6 | 4            30 | 25          42 | 18
     -----             -----            -----
   2 | 1      ,      5 |  1   ,       6 |  2    ,
and also:
  42 | 12          420 | 100
     -----             -----
   6 |  3     ,     20 |   4  .
Some oblong numbers also satisfy a Euclidean division m = d*q + r with q < r < d that are in geometric progression in this order but with a common noninteger ratio b > 1 (see A335064). (End)
For n >= 1, the continued fraction expansion of sqrt(a(n)) is [n; {2, 2n}]. For n=1, this collapses to [1; {2}]. - Magus K. Chu, Sep 09 2022
a(n-2) is the maximum irregularity over all trees with n vertices. The extremal graphs are stars. (The irregularity of a graph is the sum of the differences between the degrees over all edges of the graph.) - Allan Bickle, May 29 2023
For n > 0, number of diagonals in a regular 2*(n+1)-gon that are not parallel to any edge (cf. A367204). - Paolo Xausa, Mar 30 2024
a(n-1) is the maximum Zagreb index over all trees with n vertices. The extremal graphs are stars. (The Zagreb index of a graph is the sum of the squares of the degrees over all vertices of the graph.) - Allan Bickle, Apr 11 2024
For n >= 1, a(n) is the determinant of the distance matrix of a cycle graph on 2*n + 1 vertices (if the length of the cycle is even such a determinant is zero). - Miquel A. Fiol, Aug 20 2024
For n > 1, the continued fraction expansion of sqrt(16*a(n)) is [2n+1; {1, 2n-1, 1, 8n+2}]. - Magus K. Chu, Nov 20 2024
For n>=2, a(n) is the number of faces on a n+1-zone rhombic zonohedron. Each pair of a collection of great circles on a sphere intersects at two points, so there are 2*binomial(n+1,2) intersections. The dual of the implied polyhedron is a rhombic zonohedron, its faces corresponding to the intersections. - Shel Kaphan, Aug 12 2025

			a(3) = 12, since 2(3)+2 = 8 has 4 partitions with exactly two parts: (7,1), (6,2), (5,3), (4,4). Taking the positive differences of the parts in each partition and adding, we get: 6 + 4 + 2 + 0 = 12. - _Wesley Ivan Hurt_, Jun 02 2013
G.f. = 2*x + 6*x^2 + 12*x^3 + 20*x^4 + 30*x^5 + 42*x^6 + 56*x^7 + ... - _Michael Somos_, May 22 2014
From _Miquel Cerda_, Dec 04 2016: (Start)
a(1) = 2, since 45-43 = 2;
a(2) = 6, since 47-45 = 2 and 47-43 = 4, then 2+4 = 6;
a(3) = 12, since 49-47 = 2, 49-45 = 4, and 49-43 = 6, then 2+4+6 = 12. (End)

W. W. Berman and D. E. Smith, A Brief History of Mathematics, 1910, Open Court, page 67.
J. H. Conway and R. K. Guy, The Book of Numbers, 1996, p. 34.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
L. E. Dickson, History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 357, 1952.
L. E. Dickson, History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Chelsea, pp. 6, 232-233, 350 and 407, 1952.
H. Eves, An Introduction to the History of Mathematics, revised, Holt, Rinehart and Winston, 1964, page 72.
Nicomachus of Gerasa, Introduction to Arithmetic, translation by Martin Luther D'Ooge, Ann Arbor, University of Michigan Press, 1938, p. 254.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.6 Figurate Numbers, p. 291.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968), pp. 980-981.
C. S. Ogilvy and J. T. Anderson, Excursions in Number Theory, Oxford University Press, 1966, pp. 61-62.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 54-55.
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).
F. J. Swetz, From Five Fingers to Infinity, Open Court, 1994, p. 219.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 2-6.

T. D. Noe, Table of n, a(n) for n = 0..1000
D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, J. Int. Seq. 14 (2011) # 11.9.8.
R. Bapat, S. J. Kirkland, and M. Neumann, On distance matrices and Laplacians, Linear Algebra Appl. 401 (2005), 193-209.
Allan Bickle and Zhongyuan Che, Irregularities of Maximal k-degenerate Graphs, Discrete Applied Math. 331 (2023) 70-87.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
Allan Bickle, Zagreb Indices of Maximal k-degenerate Graphs, Australas. J. Combin. 89 1 (2024) 167-178.
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
H. Bottomley, Illustration of initial terms of A000217, A002378.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 410
P. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, Electron. J. Combin. 13 (2006), #R85, p. 11.
S. Crowley, Two new zeta constants: fractal string and hypergeometric aspects of the Riemann zeta function, viXra:1202.0066, 2012.
J. Estes and B. Wei, Sharp bounds of the Zagreb indices of k-trees, J Comb Optim 27 (2014), 271-291.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
I. Gutman and K. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
L. B. W. Jolley, Summation of Series, Dover, 1961.
Refik Keskin and Olcay Karaatli, Some New Properties of Balancing Numbers and Square Triangular Numbers, Journal of Integer Sequences, Vol. 15 (2012), Article #12.1.4.
Craig Knecht, Largest pond by area in a square.
Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 27.
Enrique Navarrete and Daniel Orellana, Finding Prime Numbers as Fixed Points of Sequences, arXiv:1907.10023 [math.NT], 2019.
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article #18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Lee Melvin Peralta, Solutions to the Equation [x]x = n, The Mathematics Teacher, Vol. 111, No. 2 (October 2017), pp. 150-154.
Aleksandar Petojević, A Note about the Pochhammer Symbol, Mathematica Moravica, Vol. 12-1 (2008), 37-42.
A. Petojevic and N. Dapic, The vAm(a,b,c;z) function, Preprint 2013.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
John D. Roth, David A. Garren, and R. Clark Robertson, Integer Carrier Frequency Offset Estimation in OFDM with Zadoff-Chu Sequences, IEEE Int'l Conference on Acoustics, Speech and Signal Processing (ICASSP 2021) 4850-4854.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
Michelle Rudolph-Lilith, On the Product Representation of Number Sequences, with Application to the Fibonacci Family, arXiv preprint arXiv:1508.07894 [math.NT], 2015.
Amelia Carolina Sparavigna, Groupoids of OEIS A002378 and A016754 Numbers (oblong and odd square numbers), Politecnico di Torino (Italy, 2019).
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
J. Striker and N. Williams, Promotion and Rowmotion , arXiv preprint arXiv:1108.1172 [math.CO], 2011-2012.
D. Suprijanto and Rusliansyah, Observation on Sums of Powers of Integers Divisible by Four, Applied Mathematical Sciences, Vol. 8, 2014, no. 45, 2219 - 2226.
Leo Tavares, Square illustration
R. Tijdeman, Some applications of Diophantine approximation, pp. 261-284 of Surveys in Number Theory (Urbana, May 21, 2000), ed. M. A. Bennett et al., Peters, 2003.
G. Villemin's Almanach of Numbers, Nombres Proniques.
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Graph Cycle.
Eric Weisstein's World of Mathematics, Pronic Number.
Eric Weisstein's World of Mathematics, Leibniz Harmonic Triangle.
Eric Weisstein's World of Mathematics, Wheel Graph.
Eric Weisstein's World of Mathematics, Wiener Index.
Wikipedia, Pronic number.
Wolfram Research, Hypergeometric Function 3F2, The Wolfram Functions site.
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Partial sums of A005843 (even numbers). Twice triangular numbers (A000217).
1/beta(n, 2) in A061928.
Cf. A035106, A087811, A119462, A127235, A049598, A124080, A033996, A028896, A046092, A000217, A005563, A046092, A001082, A059300, A059297, A059298, A166373, A002943 (bisection), A002939 (bisection), A078358 (complement).
A036689 and A036690 are subsequences. Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488. - Bruno Berselli, Jun 10 2013
Row n=2 of A185651.
Cf. A007745, A169810, A213541, A005369 (characteristic function).
Cf. A281026. - Bruno Berselli, Jan 16 2017
Cf. A045943 (4-cycles in triangular honeycomb acute knight graph), A028896 (5-cycles), A152773 (6-cycles).
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, A002943 = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
A335064 is a subsequence.
Second column of A003506.
Cf. A002378, A046092, A028896 (irregularities of maximal k-degenerate graphs).
Cf. A347213 (Dgf at s=4).
Cf. A002378, A152811, A371912 (Zagreb indices of maximal k-degenerate graphs).
Cf. A016742, A049450.

a002378 n = n * (n + 1)
a002378_list = zipWith (*) [0..] [1..]
-- Reinhard Zumkeller, Aug 27 2012, Oct 12 2011

[n*(n+1) : n in [0..100]]; // Wesley Ivan Hurt, Oct 26 2015

A002378 := proc(n)
        n*(n+1) ;
end proc:
seq(A002378(n),n=0..100) ;

Table[n(n + 1), {n, 0, 50}] (* Robert G. Wilson v, Jun 19 2004 *)
oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; Select[Range[0, 2600], oblongQ] (* Robert G. Wilson v, Sep 29 2011 *)
2 Accumulate[Range[0, 50]] (* Harvey P. Dale, Nov 11 2011 *)
LinearRecurrence[{3, -3, 1}, {2, 6, 12}, {0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)

PARI
```
{a(n) = n*(n+1)};
```

concat(0, Vec(2*x/(1-x)^3 + O(x^100))) \\ Altug Alkan, Oct 26 2015

is(n)=my(m=sqrtint(n)); m*(m+1)==n \\ Charles R Greathouse IV, Nov 01 2018

is(n)=issquare(4*n+1) \\ Charles R Greathouse IV, Mar 16 2022

def a(n): return n*(n+1)
print([a(n) for n in range(51)]) # Michael S. Branicky, Jan 13 2022

(2 to 100 by 2).scanLeft(0)( + ) // Alonso del Arte, Sep 12 2019

G.f.: 2*x/(1-x)^3. - Simon Plouffe in his 1992 dissertation.
a(n) = a(n-1) + 2*n, a(0) = 0.
Sum_{n >= 1} a(n) = n*(n+1)*(n+2)/3 (cf. A007290, partial sums).
Sum_{n >= 1} 1/a(n) = 1. (Cf. Tijdeman)
Sum_{n >= 1} (-1)^(n+1)/a(n) = log(4) - 1 = A016627 - 1 [Jolley eq (235)].
1 = 1/2 + Sum_{n >= 1} 1/(2*a(n)) = 1/2 + 1/4 + 1/12 + 1/24 + 1/40 + 1/60 + ... with partial sums: 1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, ... - Gary W. Adamson, Jun 16 2003
a(n)*a(n+1) = a(n*(n+2)); e.g., a(3)*a(4) = 12*20 = 240 = a(3*5). - Charlie Marion, Dec 29 2003
Sum_{k = 1..n} 1/a(k) = n/(n+1). - Robert G. Wilson v, Feb 04 2005
a(n) = A046092(n)/2. - Zerinvary Lajos, Jan 08 2006
Log 2 = Sum_{n >= 0} 1/a(2n+1) = 1/2 + 1/12 + 1/30 + 1/56 + 1/90 + ... = (1 - 1/2) + (1/3 - 1/4) + (1/5 - 1/6) + (1/7 - 1/8) + ... = Sum_{n >= 0} (-1)^n/(n+1) = A002162. - Gary W. Adamson, Jun 22 2003
a(n) = A110660(2*n). - N. J. A. Sloane, Sep 21 2005
a(n-1) = n^2 - n = A000290(n) - A000027(n) for n >= 1. a(n) is the inverse (frequency distribution) sequence of A000194(n). - Mohammad K. Azarian, Jul 26 2007
(2, 6, 12, 20, 30, ...) = binomial transform of (2, 4, 2). - Gary W. Adamson, Nov 28 2007
a(n) = 2*Sum_{i=0..n} i = 2*A000217(n). - Artur Jasinski, Jan 09 2007, and Omar E. Pol, May 14 2008
a(n) = A006503(n) - A000292(n). - Reinhard Zumkeller, Sep 24 2008
a(n) = A061037(4*n) = (n+1/2)^2 - 1/4 = ((2n+1)^2 - 1)/4 = (A005408(n)^2 - 1)/4. - Paul Curtz, Oct 03 2008 and Klaus Purath, Jan 13 2022
a(0) = 0, a(n) = a(n-1) + 1 + floor(x), where x is the minimal positive solution to fract(sqrt(a(n-1) + 1 + x)) = 1/2. - Hieronymus Fischer, Dec 31 2008
E.g.f.: (x+2)*x*exp(x). - Geoffrey Critzer, Feb 06 2009
Product_{i >= 2} (1-1/a(i)) = -2*sin(Pi*A001622)/Pi = -2*sin(A094886)/A000796 = 2*A146481. - R. J. Mathar, Mar 12 2009, Mar 15 2009
E.g.f.: ((-x+1)*log(-x+1)+x)/x^2 also Integral_{x = 0..1} ((-x+1)*log(-x+1) + x)/x^2 = zeta(2) - 1. - Stephen Crowley, Jul 11 2009
a(A007018(n)) = A007018(n+1), i.e., A007018(n+1) = A007018(n)-th oblong numbers. - Jaroslav Krizek, Sep 13 2009
a(n) = floor((n + 1/2)^2). a(n) = A035608(n) + A004526(n+1). - Reinhard Zumkeller, Jan 27 2010
a(n) = 2*(2*A006578(n) - A035608(n)). - Reinhard Zumkeller, Feb 07 2010
a(n-1) = floor(n^5/(n^3 + n^2 + 1)). - Gary Detlefs, Feb 11 2010
For n > 1: a(n) = A173333(n+1, n-1). - Reinhard Zumkeller, Feb 19 2010
a(n) = A004202(A000217(n)). - Reinhard Zumkeller, Feb 12 2011
a(n) = A188652(2*n+1) + 1. - Reinhard Zumkeller, Apr 13 2011
For n > 0 a(n) = 1/(Integral_{x=0..Pi/2} 2*(sin(x))^(2*n-1)*(cos(x))^3). - Francesco Daddi, Aug 02 2011
a(n) = A002061(n+1) - 1. - Omar E. Pol, Oct 03 2011
a(0) = 0, a(n) = A005408(A034856(n)) - A005408(n-1). - Ivan N. Ianakiev, Dec 06 2012
a(n) = A005408(A000096(n)) - A005408(n). - Ivan N. Ianakiev, Dec 07 2012
a(n) = A001318(n) + A085787(n). - Omar E. Pol, Jan 11 2013
Sum_{n >= 1} 1/(a(n))^(2s) = Sum_{t = 1..2*s} binomial(4*s - t - 1, 2*s - 1) * ( (1 + (-1)^t)*zeta(t) - 1). See Arxiv:1301.6293. - R. J. Mathar, Feb 03 2013
a(n)^2 + a(n+1)^2 = 2 * a((n+1)^2), for n > 0. - Ivan N. Ianakiev, Apr 08 2013
a(n) = floor(n^2 * e^(1/n)) and a(n-1) = floor(n^2 / e^(1/n)). - Richard R. Forberg, Jun 22 2013
a(n) = 2*C(n+1, 2), for n >= 0. - Felix P. Muga II, Mar 11 2014
A005369(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
Binomial transform of [0, 2, 2, 0, 0, 0, ...]. - Alois P. Heinz, Mar 10 2015
a(2n) = A002943(n) for n >= 0, a(2n-1) = A002939(n) for n >= 1. - M. F. Hasler, Oct 11 2015
For n > 0, a(n) = 1/(Integral_{x=0..1} (x^(n-1) - x^n) dx). - Rick L. Shepherd, Oct 26 2015
a(n) = A005902(n) - A007588(n). - Peter M. Chema, Jan 09 2016
For n > 0, a(n) = lim_{m -> oo} (1/m)*1/(Sum_{i=m*n..m*(n+1)} 1/i^2), with error of ~1/m. - Richard R. Forberg, Jul 27 2016
From Ilya Gutkovskiy, Jul 28 2016: (Start)
Dirichlet g.f.: zeta(s-2) + zeta(s-1).
Convolution of nonnegative integers (A001477) and constant sequence (A007395).
Sum_{n >= 0} a(n)/n! = 3*exp(1). (End)
From Charlie Marion, Mar 06 2020: (Start)
a(n)*a(n+2k-1) + (n+k)^2 = ((2n+1)*k + n^2)^2.
a(n)*a(n+2k) + k^2 = ((2n+1)*k + a(n))^2. (End)
Product_{n>=1} (1 + 1/a(n)) = cosh(sqrt(3)*Pi/2)/Pi. - Amiram Eldar, Jan 20 2021
A generalization of the Dec 29 2003 formula, a(n)*a(n+1) = a(n*(n+2)), follows. a(n)*a(n+k) = a(n*(n+k+1)) + (k-1)*n*(n+k+1). - Charlie Marion, Jan 02 2023
a(n) = A016742(n) - A049450(n). - Leo Tavares, Mar 15 2025

Additional comments from Michael Somos

Comment and cross-reference added by Christopher Hunt Gribble, Oct 13 2009

A005843 The nonnegative even numbers: a(n) = 2n.

Original entry on oeis.org

0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, 116, 118, 120

Offset: 0

N. J. A. Sloane

-2, -4, -6, -8, -10, -12, -14, ... are the trivial zeros of the Riemann zeta function. - Vivek Suri (vsuri(AT)jhu.edu), Jan 24 2008
If a 2-set Y and an (n-2)-set Z are disjoint subsets of an n-set X then a(n-2) is the number of 2-subsets of X intersecting both Y and Z. - Milan Janjic, Sep 19 2007
A134452(a(n)) = 0; A134451(a(n)) = 2 for n > 0. - Reinhard Zumkeller, Oct 27 2007
Omitting the initial zero gives the number of prime divisors with multiplicity of product of terms of n-th row of A077553. - Ray Chandler, Aug 21 2003
A059841(a(n))=1, A000035(a(n))=0. - Reinhard Zumkeller, Sep 29 2008
(APSO) Alternating partial sums of (a-b+c-d+e-f+g...) = (a+b+c+d+e+f+g...) - 2*(b+d+f...), it appears that APSO(A005843) = A052928 = A002378 - 2*(A116471), with A116471=2*A008794. - Eric Desbiaux, Oct 28 2008
A056753(a(n)) = 1. - Reinhard Zumkeller, Aug 23 2009
Twice the nonnegative numbers. - Juri-Stepan Gerasimov, Dec 12 2009
The number of hydrogen atoms in straight-chain (C(n)H(2n+2)), branched (C(n)H(2n+2), n > 3), and cyclic, n-carbon alkanes (C(n)H(2n), n > 2). - Paul Muljadi, Feb 18 2010
For n >= 1; a(n) = the smallest numbers m with the number of steps n of iterations of {r - (smallest prime divisor of r)} needed to reach 0 starting at r = m. See A175126 and A175127. A175126(a(n)) = A175126(A175127(n)) = n. Example (a(4)=8): 8-2=6, 6-2=4, 4-2=2, 2-2=0; iterations has 4 steps and number 8 is the smallest number with such result. - Jaroslav Krizek, Feb 15 2010
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is not integer. A040001(a(n)) > 1. See A145051 and A040001. - Jaroslav Krizek, May 28 2010
Union of A179082 and A179083. - Reinhard Zumkeller, Jun 28 2010
a(k) is the (Moore lower bound on and the) order of the (k,4)-cage: the smallest k-regular graph having girth four: the complete bipartite graph with k vertices in each part. - Jason Kimberley, Oct 30 2011
For n > 0: A048272(a(n)) <= 0. - Reinhard Zumkeller, Jan 21 2012
Let n be the number of pancakes that have to be divided equally between n+1 children. a(n) is the minimal number of radial cuts needed to accomplish the task. - Ivan N. Ianakiev, Sep 18 2013
For n > 0, a(n) is the largest number k such that (k!-n)/(k-n) is an integer. - Derek Orr, Jul 02 2014
a(n) when n > 2 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
It appears that for n > 2, a(n) = A020482(n) + A002373(n), where all sequences are infinite. This is consistent with Goldbach's conjecture, which states that every even number > 2 can be expressed as the sum of two prime numbers. - Bob Selcoe, Mar 08 2015
Number of partitions of 4n into exactly 2 parts. - Colin Barker, Mar 23 2015
Number of neighbors in von Neumann neighborhood. - Dmitry Zaitsev, Nov 30 2015
Unique solution b( ) of the complementary equation a(n) = a(n-1)^2 - a(n-2)*b(n-1), where a(0) = 1, a(1) = 3, and a( ) and b( ) are increasing complementary sequences. - Clark Kimberling, Nov 21 2017
Also the maximum number of non-attacking bishops on an (n+1) X (n+1) board (n>0). (Cf. A000027 for rooks and queens (n>3), A008794 for kings or A030978 for knights.) - Martin Renner, Jan 26 2020
Integer k is even positive iff phi(2k) > phi(k), where phi is Euler's totient (A000010) [see reference De Koninck & Mercier]. - Bernard Schott, Dec 10 2020
Number of 3-permutations of n elements avoiding the patterns 132, 213, 312 and also number of 3-permutations avoiding the patterns 213, 231, 321. See Bonichon and Sun. - Michel Marcus, Aug 20 2022
a(n) gives the y-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The x-value is given by 2*n^2 + 1 (see Tattersall). - Stefano Spezia, Jul 24 2025

			G.f. = 2*x + 4*x^2 + 6*x^3 + 8*x^4 + 10*x^5 + 12*x^6 + 14*x^7 + 16*x^8 + ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 2.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 28.
J.-M. De Koninck and A. Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 529a pp. 71 and 257, Ellipses, 2004, Paris.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Nicolas Bonichon and Pierre-Jean Morel, Baxter d-permutations and other pattern avoiding classes, arXiv:2202.12677 [math.CO], 2022.
David Callan, On Ascent, Repetition and Descent Sequences, arXiv:1911.02209 [math.CO], 2019.
Charles Cratty, Samuel Erickson, Frehiwet Negass, and Lara Pudwell, Pattern Avoidance in Double Lists, preprint, 2015.
Kevin Fagan, Drabble cartoon, Jun 15 1987: Intelligence Test
Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012, J. Int. Seq. 15 (2012) # 12.6.2
Milan Janjic, Two Enumerative Functions
Tanya Khovanova, Recursive Sequences
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Nathan Sun, On d-permutations and Pattern Avoidance Classes, arXiv:2208.08506 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Even Number
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Riemann Zeta Function Zeros
Wikipedia, Alkane
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (2,-1).

a(n)=2*A001477(n). - Juri-Stepan Gerasimov, Dec 12 2009
Cf. A000027, A002061, A005408, A001358, A077553, A077554, A077555, A002024, A087112, A157888, A157889, A140811, A157872, A157909, A157910, A165900.
Moore lower bound on the order of a (k,g) cage: A198300 (square); rows: A000027 (k=2), A027383 (k=3), A062318 (k=4), A061547 (k=5), A198306 (k=6), A198307 (k=7), A198308 (k=8), A198309 (k=9), A198310 (k=10), A094626 (k=11); columns: A020725 (g=3), this sequence (g=4), A002522 (g=5), A051890 (g=6), A188377 (g=7). - Jason Kimberley, Oct 30 2011
Cf. A231200 (boustrophedon transform).

a005843 = (* 2)
a005843_list = [0, 2 ..]  -- Reinhard Zumkeller, Feb 11 2012

Magma
```
[ 2*n : n in [0..100]];
```

A005843 := n->2*n;
A005843:=2/(z-1)**2; # Simon Plouffe in his 1992 dissertation

Range[0,120,2] (* Harvey P. Dale, Aug 16 2011 *)

PARI
```
A005843(n) = 2*n
```

def a(n): return 2*n # Martin Gergov, Oct 20 2022

R
```
seq(0,200,2)
```

G.f.: 2*x/(1-x)^2.
E.g.f.: 2*x*exp(x). - Geoffrey Critzer, Aug 25 2012
G.f. with interpolated zeros: 2x^2/((1-x)^2 * (1+x)^2); e.g.f. with interpolated zeros: x*sinh(x). - Geoffrey Critzer, Aug 25 2012
Inverse binomial transform of A036289, n*2^n. - Joshua Zucker, Jan 13 2006
a(0) = 0, a(1) = 2, a(n) = 2a(n-1) - a(n-2). - Jaume Oliver Lafont, May 07 2008
a(n) = Sum_{k=1..n} floor(6n/4^k + 1/2). - Vladimir Shevelev, Jun 04 2009
a(n) = A034856(n+1) - A000124(n) = A000217(n) + A005408(n) - A000124(n) = A005408(n) - 1. - Jaroslav Krizek, Sep 05 2009
a(n) = Sum_{k>=0} A030308(n,k)*A000079(k+1). - Philippe Deléham, Oct 17 2011
Digit sequence 22 read in base n-1. - Jason Kimberley, Oct 30 2011
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Dec 23 2011
a(n) = 2*n = Product_{k=1..2*n-1} 2*sin(Pi*k/(2*n)), n >= 0 (undefined product := 1). See an Oct 09 2013 formula contribution in A000027 with a reference. - Wolfdieter Lang, Oct 10 2013
From Ilya Gutkovskiy, Aug 19 2016: (Start)
Convolution of A007395 and A057427.
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2)/2 = (1/2)*A002162 = (1/10)*A016655. (End)
From Bernard Schott, Dec 10 2020: (Start)
Sum_{n>=1} 1/a(n)^2 = Pi^2/24 = A222171.
Sum_{n>=1} (-1)^(n+1)/a(n)^2 = Pi^2/48 = A245058. (End)

A000004 The zero sequence.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, Table of n, a(n) for n = 0..1000 [Useful when plotting one sequence against another.]
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 10.
Index entries for characteristic functions
Index entries for "core" sequences
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for sequences that are fixed points of mappings

Cf. A000012 (all 1's), A007395 (all 2's), A010701 (all 3's).

Cf. A000007(n) = 0^n: characteristic function of {0}.

a000004 = const 0
a000004_list = repeat 0  -- Reinhard Zumkeller, May 07 2012

Magma
```
[ 0 : n in [0..100]];
```
Maple
```
A000004 := n->0;
```

a[ n_] := 0;
Table[0, {n, 100}] (* Matthew House, Jul 14 2015 *)
LinearRecurrence[{1},{0},102] (* Ray Chandler, Jul 15 2015 *)

PARI
```
vector(100,n,0)
```

print([0 for n in range(102)]) # Michael S. Branicky, Apr 04 2022

R
```
rep(0,100)
```

a(n) = 0 for all integer n.

A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 7, 9, 5, 1, 1, 9, 16, 14, 6, 1, 1, 11, 25, 30, 20, 7, 1, 1, 13, 36, 55, 50, 27, 8, 1, 1, 15, 49, 91, 105, 77, 35, 9, 1, 1, 17, 64, 140, 196, 182, 112, 44, 10, 1, 1, 19, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 21, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1

Offset: 1

Clark Kimberling, Feb 28 2012

Row sums:  A083329
Alternating row sums:  1,0,-1,-1,-1,-1,-1,-1,-1,-1,...
Antidiagonal sums: A000071 (-1+Fibonacci numbers)
col 1:  A000012
col 2:  A005408
col 3:  A000290
col 4:  A000330
col 5:  A002415
col 6:  A005585
col 7:  A040977
col 8:  A050486
col 9:  A053347
col 10:  A054333
col 11:  A054334
col 12:  A057788
col 2n-1 of A208510 is column n of A208508
col 2n of A208510 is column n of A208509.
...
GENERAL DISCUSSION:
A208510 typifies arrays generated by paired recurrence equations of the following form:
u(n,x)=a(n,x)*u(n-1,x)+b(n,x)*v(n-1,x)+c(n,x)
v(n,x)=d(n,x)*u(n-1,x)+e(n,x)*v(n-1,x)+f(n,x).
...
These first-order recurrences imply separate second-order recurrences.  In order to show them, the six functions a(n,x),...,f(n,x) are abbreviated as a,b,c,d,e,f.
Then, starting with initial values u(1,x)=1 and u(2,x)=a+b+c: u(n,x) = (a+e)u(n-1,x) + (bd-ae)u(n-2,x) + bf-ce+c.
With initial values v(1,x)=1 and v(2,x)=d+e+f: v(n,x) = (a+e)v(n-1,x) + (bd-ae)v(n-2,x) + cd-af+f.
...
In the guide below, the last column codes certain sequences that occur in one of these ways: row, column, edge, row sum, alternating row sum.  Coding:
A: 1,-1,1,-1,1,-1,1.... A033999
B: 1,2,4,8,16,32,64,... powers of 2
C: 1,1,1,1,1,1,1,1,.... A000012
D: 2,2,2,2,2,2,2,2,.... A007395
E: 2,4,6,8,10,12,14,... even numbers
F: 1,1,2,3,5,8,13,21,.. Fibonacci numbers
N: 1,2,3,4,5,6,7,8,.... A000027
O: 1,3,5,7,9,11,13,.... odd numbers
P: 1,3,9,27,81,243,.... powers of 3
S: 1,4,9,16,25,36,49,.. squares
T: 1,3,6,10,15,21,38,.. triangular numbers
Z: 1,0,0,0,0,0,0,0,0,.. A000007
*: (eventually) periodic alternating row sums
^: has a limiting row; i.e., the polynomials "approach" a power series
This coding includes indirect and repeated occurrences; e.g. F occurs thrice at A094441: in column 1 directly as Fibonacci numbers, in row sums as odd-indexed Fibonacci numbers, and in alternating row sums as signed Fibonacci numbers.
......... a....b....c....d....e....f....code
A034839 u 1....1....0....1....x....0....CCOT
A034867 v 1....1....0....1....x....0....CEN
A210221 u 1....1....0....1....2x...0....BBFF
A210596 v 1....1....0....1....2x...0....BBFF
A105070 v 1....2x...0....1....1....0....BN
A207605 u 1....1....0....1....x+1..0....BCFFN
A106195 v 1....1....0....1....x+1..0....BCFFN
A207606 u 1....1....0....x....x+1..0....DNT
A207607 v 1....1....0....x....x+1..0....DNT
A207608 u 1....1....0....2x...x+1..0....N
A207609 v 1....1....0....2x...x+1..0....C
A207610 u 1....1....0....1....x....1....CF
A207611 v 1....1....0....1....x....1....BCF
A207612 u 1....1....0....1....2x...1....BF
A207613 v 1....1....0....1....2x...1....BF
A207614 u 1....1....0....1....x+1..1....CN
A207615 v 1....1....0....1....x+1..1....CFN
A207616 u 1....1....0....x....1....1....CE
A207617 v 1....1....0....x....1....1....CNO
A029638 u 1....1....0....x....x....1....CDNO
A029635 v 1....1....0....x....x....1....CDNOZ
A207618 u 1....1....0....x....2x...1....N
A207619 v 1....1....0....x....2x...1....CFN
A207620 u 1....1....0....x....x+1..1....DET
A207621 v 1....1....0....x....x+1..1....DNO
A207622 u 1....1....0....2x...1....1....BT
A207623 v 1....1....0....2x...1....1....BN
A207624 u 1....1....0....2x...x....1....N
A102662 v 1....1....0....2x...x....1....CO
A207625 u 1....1....0....2x...x+1..1....T
A207626 v 1....1....0....2x...x+1..1....N
A207627 u 1....1....0....2x...2x...1....BN
A207628 v 1....1....0....2x...2x...1....BCE
A207629 u 1....1....0....x+1..1....1....CET
A207630 v 1....1....0....x+1..1....1....CO
A207631 u 1....1....0....x+1..x....1....DF
A207632 v 1....1....0....x+1..x....1....DEF
A207633 u 1....1....0....x+1..2x...1....F
A207634 v 1....1....0....x+1..2x...1....F
A207635 u 1....1....0....x+1..x+1..1....DN
A207636 v 1....1....0....x+1..x+1..1....CD
A160232 u 1....x....0....1....2x...0....BCFN
A208341 v 1....x....0....1....2x...0....BCFFN
A085478 u 1....x....0....1....x+1..0....CCOFT*
A078812 v 1....x....0....1....x+1..0....CEFN*
A208342 u 1....x....0....x....x....0....CCFNO
A208343 v 1....x....0....x....x....0....BBCDFZ
A208344 u 1....x....0....x....2x...0....CCFN
A208345 v 1....x....0....x....2x...0....CFZ
A094436 u 1....x....0....x....x+1..0....CFFN
A094437 v 1....x....0....x....x+1..0....CEFF
A117919 u 1....x....0....2x...1....0....BCNT
A135837 v 1....x....0....2x...1....0....BCET
A208328 u 1....x....0....2x...x....0....CCOP
A208329 v 1....x....0....2x...x....0....DPZ
A208330 u 1....x....0....2x...x+1..0....CNPT
A208331 v 1....x....0....2x...x+1..0....CN
A208332 u 1....x....0....2x...2x...0....CCE
A208333 v 1....x....0....2x...2x...0....DZ
A208334 u 1....x....0....x+1..1....0....CCNT
A208335 v 1....x....0....x+1..1....0....CCN*
A208336 u 1....x....0....x+1..x....0....CFNT*
A208337 v 1....x....0....x+1..x....0....ACFN*
A208338 u 1....x....0....x+1..2x...0....CNP
A208339 v 1....x....0....x+1..2x...0....BCNP
A202390 u 1....x....0....x+1..x+1..0....CFPTZ*
A208340 v 1....x....0....x+1..x+1..0....FNPZ*
A208508 u 1....x....0....1....1....1....CCES
A208509 v 1....x....0....1....1....1....BCO
A208510 u 1....x....0....1....x....1....CCCNOS*
A029653 v 1....x....0....1....x....1....BCDOSZ*
A208511 u 1....x....0....1....2x...1....BCFO
A208512 v 1....x....0....1....2x...1....BDFO
A208513 u 1....x....0....1....x+1..1....CCES*
A111125 v 1....x....0....1....x+1..1....COO*
A133567 u 1....x....0....x....1....1....CCOTT
A133084 v 1....x....0....x....1....1....BBCEN
A208514 u 1....x....0....x....x....1....CEFN
A208515 v 1....x....0....x....x....1....BCDFN
A208516 u 1....x....0....x....2x...1....CNN
A208517 v 1....x....0....x....2x...1....CCN
A208518 u 1....x....0....x....x+1..1....CFNT
A208519 v 1....x....0....x....x+1..1....NFFT
A208520 u 1....x....0....2x...1....1....BCTT
A208521 v 1....x....0....2x...1....1....BEN
A208522 u 1....x....0....2x...x....1....CCN
A208523 v 1....x....0....2x...x....1....CCO
A208524 u 1....x....0....2x...x+1..1....CT*
A208525 v 1....x....0....2x...x+1..1....ACNP*
A208526 u 1....x....0....2x...2x...1....CEN
A208527 v 1....x....0....2x...2x...1....CCE
A208606 u 1....x....0....x+1..1....1....CCS
A208607 v 1....x....0....x+1..1....1....CNO
A208608 u 1....x....0....x+1..x....1....CFOT
A208609 v 1....x....0....x+1..x....1....DEN*
A208610 u 1....x....0....x+1..2x...1....CO
A208611 v 1....x....0....x+1..2x...1....DE
A208612 u 1....x....0....x+1..x+1..1....CFNS
A208613 v 1....x....0....x+1..x+1..1....CFN*
A105070 u 1....2x...0....1....1....0....BN
A207536 u 1....2x...0....1....1....0....BCT
A208751 u 1....2x...0....1....x+1..0....CDPT
A208752 v 1....2x...0....1....x+1..0....CNP
A135837 u 1....2x...0....x....1....0....BCNT
A117919 v 1....2x...0....x....1....0....BCNT
A208755 u 1....2x...0....x....x....0....BCDEP
A208756 v 1....2x...0....x....x....0....BCCOZ
A208757 u 1....2x...0....x....2x...0....CDEP
A208758 v 1....2x...0....x....2x...0....CCEPZ
A208763 u 1....2x...0....2x...x....0....CDOP
A208764 v 1....2x...0....2x...x....0....CCCP
A208765 u 1....2x...0....2x...x+1..0....CE
A208766 v 1....2x...0....2x...x+1..0....CC
A208747 u 1....2x...0....2x...2x...0....CDE
A208748 v 1....2x...0....2x...2x...0....CCZ
A208749 u 1....2x...0....x+1..1....0....BCOPT
A208750 v 1....2x...0....x+1..1....0....BCNP*
A208759 u 1....2x...0....x+1..2x....0...CE
A208760 v 1....2x...0....x+1..2x....0...BCO
A208761 u 1....2x...0....x+1..x+1...0...BCCT*
A208762 v 1....2x...0....x+1..x+1...0...BNZ*
A208753 u 1....2x...0....1....1.....1...BCS
A208754 v 1....2x...0....1....1.....1...BO
A105045 u 1....2x...0....1....2x....1...BCCOS*
A208659 v 1....2x...0....1....2x....1...BDOSZ*
A208660 u 1....2x...0....1....x+1...1...CDS
A208904 v 1....2x...0....1....x+1...1...CNO
A208905 u 1....2x...0....x....1.....1...BCT
A208906 v 1....2x...0....x....1.....1...BNN
A208907 u 1....2x...0....x....x.....1...BCN
A208756 v 1....2x...0....x....x.....1...BCCE
A208755 u 1....2x...0....x....2x....1...CEN
A208910 v 1....2x...0....x....2x....1...CCE
A208911 u 1....2x...0....x....x+1...1...BCT
A208912 v 1....2x...0....x....x+1...1...BNT
A208913 u 1....2x...0....2x...1.....1...BCT
A208914 v 1....2x...0....2x...1.....1...BEN
A208915 u 1....2x...0....2x...x.....1...CE
A208916 v 1....2x...0....2x...x.....1...CCO
A208919 u 1....2x...0....2x...x+1...1...CT
A208920 v 1....2x...0....2x...x+1...1...N
A208917 u 1....2x...0....2x...2x....1...CEN
A208918 v 1....2x...0....2x...2x....1...CCNP
A208921 u 1....2x...0....x+1..1.....1...BC
A208922 v 1....2x...0....x+1..1.....1...BON
A208923 u 1....2x...0....x+1..x.....1...BCNO
A208908 v 1....2x...0....x+1..x.....1...BDN*
A208909 u 1....2x...0....x+1..2x....1...BN
A208930 v 1....2x...0....x+1..2x....1...DN
A208931 u 1....2x...0....x+1..x+1...1...BCOS
A208932 v 1....2x...0....x+1..x+1...1...BCO*
A207537 u 1....x+1..0....1....1.....0...BCO
A207538 v 1....x+1..0....1....1.....0...BCE
A122075 u 1....x+1..0....1....x.....0...CCFN*
A037027 v 1....x+1..0....1....x.....0...CCFN*
A209125 u 1....x+1..0....1....2x....0...BCFN*
A164975 v 1....x+1..0....1....2x....0...BF
A209126 u 1....x+1..0....x....x.....0...CDFO*
A209127 v 1....x+1..0....x....x.....0...DFOZ*
A209128 u 1....x+1..0....x....2x....0...CDE*
A209129 v 1....x+1..0....x....2x....0...DEZ
A102756 u 1....x+1..0....x....x+1...0...CFNP*
A209130 v 1....x+1..0....x....x+1...0...CCFNP*
A209131 u 1....x+1..0....2x...x.....0...CDEP*
A209132 v 1....x+1..0....2x...x.....0...CNPZ*
A209133 u 1....x+1..0....2x...2x....0...CDN
A209134 v 1....x+1..0....2x...2x....0...CCN*
A209135 u 1....x+1..0....2x...x+1...0...CN*
A209136 v 1....x+1..0....2x...x+1...0...CCS*
A209137 u 1....x+1..0....x+1..x.....0...CFFP*
A209138 v 1....x+1..0....x+1..x.....0...AFFP*
A209139 u 1....x+1..0....x+1..2x....0...CF*
A209140 v 1....x+1..0....x+1..2x....0...BF
A209141 u 1....x+1..0....x+1..x+1...0...BCF*
A209142 v 1....x+1..0....x+1..x+1...0...BFZ*
A209143 u 1....x+1..0....1....1.....1...CCE*
A209144 v 1....x+1..0....1....1.....1...COO*
A209145 u 1....x+1..0....1....x.....1...CCFN*
A122075 v 1....x+1..0....1....x.....1...CCFN*
A209146 u 1....x+1..0....1....2x....1...BCF*
A209147 v 1....x+1..0....1....2x....1...BF
A209148 u 1....x+1..0....1....x+1...1...CCO*
A209149 v 1....x+1..0....1....x+1...1...CDO*
A209150 u 1....x+1..0....x....1.....1...CCNT*
A208335 v 1....x+1..0....x....1.....1...CDNN*
A209151 u 1....x+1..0....x....x.....1...CFN*
A208337 v 1....x+1..0....x....x.....1...ACFN*
A209152 u 1....x+1..0....x....2x....1...CN*
A208339 v 1....x+1..0....x....x.....1...BCN
A209153 u 1....x+1..0....x....x+1...1...CFT*
A208340 v 1....x+1..0....x....x.....1...FNZ*
A209154 u 1....x+1..0....2x...1.....1...BCT*
A209157 v 1....x+1..0....2x...1.....1...BNN
A209158 u 1....x+1..0....2x...x.....1...CN*
A209159 v 1....x+1..0....2x...x.....1...CO*
A209160 u 1....x+1..0....2x...2x....1...CN*
A209161 v 1....x+1..0....2x...2x....1...CE
A209162 u 1....x+1..0....2x...x+1...1...CT*
A209163 v 1....x+1..0....2x...x+1...1...CO*
A209164 u 1....x+1..0....x+1..1.....1...CC*
A209165 v 1....x+1..0....x+1..1.....1...CCN
A209166 u 1....x+1..0....x+1..x.....1...CFF*
A209167 v 1....x+1..0....x+1..x.....1...FF*
A209168 u 1....x+1..0....x+1..2x....1...CF*
A209169 v 1....x+1..0....x+1..2x....1...CF
A209170 u 1....x+1..0....x+1..x+1...1...CF*
A209171 v 1....x+1..0....x+1..x+1...1...CF*
A053538 u x....1....0....1....1.....0...BBCCFN
A076791 v x....1....0....1....1.....0...BBCDF
A209172 u x....1....0....1....2x....0...BCCFF
A209413 v x....1....0....1....2x....0...BCCFF
A094441 u x....1....0....1....x+1...0...CFFFN
A094442 v x....1....0....1....x+1...0...CEFFF
A054142 u x....1....0....x....x+1...0...CCFOT*
A172431 v x....1....0....x....x+1...0...CEFN*
A008288 u x....1....0....2x...1.....0...CCOO*
A035607 v x....1....0....2x...1.....0...ACDE*
A209414 u x....1....0....2x...x+1...0...CCS
A112351 v x....1....0....2x...x+1...0...CON
A209415 u x....1....0....x+1..x.....0...CCTN
A209416 v x....1....0....x+1..x.....0...ACN*
A209417 u x....1....0....x+1..2x....0...CC
A209418 v x....1....0....x+1..2x....0...BBC
A209419 u x....1....0....x+1..x+1...0...CFTZ*
A209420 v x....1....0....x+1..x+1...0...FNZ*
A209421 u x....1....0....1....1.....1...CCN
A209422 v x....1....0....1....1.....1...CD
A209555 u x....1....0....1....x.....1...CNN
A209556 v x....1....0....1....x.....1...CNN
A209557 u x....1....0....1....2x....1...BCN
A209558 v x....1....0....1....2x....1...BN
A209559 u x....1....0....1....x+1...1...CN
A209560 v x....1....0....1....x+1...1...CN
A209561 u x....1....0....x....1.....1...CCNNT*
A209562 v x....1....0....x....1.....1...CDNNT*
A209563 u x....1....0....x....x.....1...CCFT^
A209564 v x....1....0....x....x.....1...CFN^
A209565 u x....1....0....x....2x....1...CC^
A209566 v x....1....0....x....2x....1...BC^
A209567 u x....1....0....x....x+1...1...CNT*
A209568 v x....1....0....x....x+1...1...NNS*
A209569 u x....1....0....2x...1.....1...CNO*
A209570 v x....1....0....2x...1.....1...DNN*
A209571 u x....1....0....2x...x.....1...CCS^
A209572 v x....1....0....2x...x.....1...CN^
A209573 u x....1....0....2x...x+1...1...CNS
A209574 v x....1....0....2x...x+1...1...NO
A209575 u x....1....0....2x...2x....1...CC
A209576 v x....1....0....2x...2x....1...C
A209577 u x....1....0....x+1..1.....1...CNNT
A209578 v x....1....0....x+1..1.....1...CNN
A209579 u x....1....0....x+1..x.....1...CNNT
A209580 v x....1....0....x+1..x.....1...NN*
A209581 u x....1....0....x+1..2x....1...CN
A209582 v x....1....0....x+1..2x....1...BN
A209583 u x....1....0....x+1..x+1...1...CT*
A209584 v x....1....0....x+1..x+1...1...CN*
A121462 u x....x....0....x....x+1...0...BCFFNZ
A208341 v x....x....0....x....x+1...0...BCFFN
A209687 u x....x....0....2x...x+1...0...BCNZ
A208339 v x....x....0....2x...x+1...0...BCN
A115241 u x....x....0....1....1.....1...CDNZ*
A209688 v x....x....0....1....1.....1...DDN*
A209689 u x....x....0....1....x.....1...FNZ^
A209690 v x....x....0....1....x.....1...FN^
A209691 u x....x....0....1....2x....1...BCZ^
A209692 v x....x....0....1....2x....1...BCC^
A209693 u x....x....0....1....x+1...1...NNZ*
A209694 v x....x....0....1....x+1...1...CN*
A209697 u x....x....0....x....x+1...1...BNZ
A209698 v x....x....0....x....x+1...1...BNT
A209699 u x....x....0....2x...1.....1...BNNZ
A209700 v x....x....0....2x...1.....1...BDN
A209701 u x....x....0....2x...x+1...1...NZ
A209702 v x....x....0....2x...x+1...1...N
A209703 u x....x....0....x+1..1.....1...FNTZ
A209704 v x....x....0....x+1..1.....1...FNNT
A209705 u x....x....0....x+1..x+1...1...BNZ*
A209706 v x....x....0....x+1..x+1...1...BCN*
A209695 u x....x+1..0....2x...x+1...0...ACN*
A209696 v x....x+1..0....2x...x+1...0...CDN*
A209830 u x....x+1..0....x+1..2x....0...ACF
A209831 v x....x+1..0....x+1..2x....0...BCF*
A209745 u x....x+1..0....x+1..x+1...0...ABF*
A209746 v x....x+1..0....x+1..x+1...0...BFZ*
A209747 u x....x+1..0....1....1.....1...ADE*
A209748 v x....x+1..0....1....1.....1...DEO
A209749 u x....x+1..0....1....x.....1...ANN*
A209750 v x....x+1..0....1....x.....1...CNO
A209751 u x....x+1..0....1....2x....1...ABN*
A209752 v x....x+1..0....1....2x....1...BN
A209753 u x....x+1..0....1....x+1...1...AN*
A209754 v x....x+1..0....1....x+1...1...NT*
A209755 u x....x+1..0....x....1.....1...AFN
A209756 v x....x+1..0....x....1.....1...FNO*
A209759 u x....x+1..0....x....2x....1...ACF^
A209760 v x....x+1..0....x....2x....1...CF^*
A209761 u x....x+1..0....x.....x+1..1...ABNS*
A209762 v x....x+1..0....x.....x+1..1...BNS*
A209763 u x....x+1..0....2x....1....1...ABN*
A209764 v x....x+1..0....2x....1....1...BNN
A209765 u x....x+1..0....2x....x....1...ACF^*
A209766 v x....x+1..0....2x....x....1...CF^
A209767 u x....x+1..0....2x....x+1..1...AN*
A209768 v x....x+1..0....2x....x+1..1...N*
A209769 u x....x+1..0....x+1...1....1...AF*
A209770 v x....x+1..0....x+1...1....1...FN
A209771 u x....x+1..0....x+1...x....1...ABN*
A209772 v x....x+1..0....x+1...x....1...BN*
A209773 u x....x+1..0....x+1...2x...1...AF
A209774 v x....x+1..0....x+1...2x...1...FN*
A209775 u x....x+1..0....x+1...x+1..1...AB*
A209776 v x....x+1..0....x+1...x+1..1...BC*
A210033 u 1....1....1....1.....x....1...BCN
A210034 v 1....1....1....1.....x....1...BCDFN
A210035 u 1....1....1....1.....2x...1...BBF
A210036 v 1....1....1....1.....2x...1...BBFF
A210037 u 1....1....1....1.....x+1..1...BCFFN
A210038 v 1....1....1....1.....x+1..1...BCFFN
A210039 u 1....1....1....x.....1....1...BCOT
A210040 v 1....1....1....x.....1....1...BCEN
A210042 u 1....1....1....x.....x....1...BCDEOT*
A124927 v 1....1....1....x.....x....1...BCDET*
A210041 u 1....1....1....x.....2x...1...BFO
A209758 v 1....1....1....x.....2x...1...BCFO
A210187 u 1....1....1....x.....x+1..1...DTF*
A210188 v 1....1....1....x.....x+1..1...DNF*
A210189 u 1....1....1....2x....1....1...BT
A210190 v 1....1....1....2x....1....1...BN
A210191 u 1....1....1....2x....x....1...CO*
A210192 v 1....1....1....2x....x....1...CCO*
A210193 u 1....1....1....2x....x+1..1...CPT
A210194 v 1....1....1....2x....x+1..1...CN
A210195 u 1....1....1....2x....2x...1...BOPT*
A210196 v 1....1....1....2x....2x...1...BCC*
A210197 u 1....1....1....x+1...1....1...BCOT
A210198 v 1....1....1....x+1...1....1...BCEN
A210199 u 1....1....1....x+1...x....1...DFT
A210200 v 1....1....1....x+1...x....1...DFO*
A210201 u 1....1....1....x+1...2x...1...BFP
A210202 v 1....1....1....x+1...2x...1...BF
A210203 u 1....1....1....x+1...x+1..1...BDOP
A210204 v 1....1....1....x+1...x+1..1...BCDN*
A210211 u x....1....1....1.....2x...1...BCFN
A210212 v x....1....1....1.....2x...1...BFN
A210213 u x....1....1....1.....x+1..1...CFFN
A210214 v x....1....1....1.....x+1..1...CFFO
A210215 u x....1....1....x.....x....1...BCDFT^
A210216 v x....1....1....x.....x....1...BCFO^
A210217 u x....1....1....x.....2x...1...CDF^
A210218 v x....1....1....x.....2x...1...BCF^
A210219 u x....1....1....x.....x+1..1...CNSTF*
A210220 v x....1....1....x.....x+1..1...FNNT*
A104698 u x....1....1....2x......1..1...CENS*
A210220 v x....1....1....2x....x+1..1...DNNT*
A210223 u x....1....1....2x....x....1...CD^
A210224 v x....1....1....2x....x....1...CO^
A210225 u x....1....1....2x....x+1..1...CNP
A210226 v x....1....1....2x....x+1..1...NOT
A210227 u x....1....1....2x....2x...1...CDP^
A210228 v x....1....1....2x....2x...1...C^
A210229 u x....1....1....x+1...1....1...CFNN
A210230 v x....1....1....x+1...1....1...CCN
A210231 u x....1....1....x+1...x....1...CNT
A210232 v x....1....1....x+1...x....1...NN*
A210233 u x....1....1....x+1...2x...1...CNP
A210234 v x....1....1....x+1...2x...1...BN
A210235 u x....1....1....x+1...x+1..1...CCFPT*
A210236 v x....1....1....x+1...x+1..1...CFN*
A124927 u x....x....1....1.....1....1...BCDEET*
A210042 v x....1....1....x+1...x+1..1...BDEOT*
A210216 u x....x....1....1.....x....1...BCFO^
A210215 v x....x....1....1.....x....1...BCDFT^
A210549 u x....x....1....1.....2x...1...BCF^
A210550 v x....x....1....1.....2x...1...BDF^
A172431 u x....x....1....1.....x+1..1...CEFN*
A210551 v x....x....1....1.....x+1..1...CFOT*
A210552 u x....x....1....x.....1....1...BBCFNO
A210553 v x....x....1....x.....1....1...BNNFB
A208341 u x....x....1....x.....x+1..1...BCFFN
A210554 v x....x....1....x.....x+1..1...BNFFT
A210555 u x....x....1....2x....1....1...BCNN
A210556 v x....x....1....2x....1....1...BENP
A210557 u x....x....1....2x....x+1..1...CNP
A210558 v x....x....1....2x....x+1..1...N
A210559 u x....x....1....x+1...1....1...CEF
A210560 v x....x....1....x+1...1....1...OFNS
A210561 u x....x....1....x+1...x....1...BCNP^
A210562 v x....x....1....x+1...x....1...BDP*^
A210563 u x....x....1....x+1...2x...1...CFP^
A210564 v x....x....1....x+1...2x...1...DF^
A013609 u x....x....1....x+1...x+1..1...BCEPT*
A209757 v x....x....1....x+1...x+1..1...BCOS*
A209819 u x....2x...1....x+1...x....1...CFN^
A209820 v x....2x...1....x+1...x....1...DF^
A209996 u x....2x...1....x+1...2x...1...CP^
A209998 v x....2x...1....x+1...2x...1...DP^
A209999 u x....x+1..1....1.....x+1..1...FN*
A210287 v x....x+1..1....1.....x+1..1...CFT*
A210565 u x....x+1..1....x.....1....1...FNT*
A210595 v x....x+1..1....x.....1....1...FNNT
A210598 u x....x+1..1....x+1...2x...1...FN*
A210599 v x....x+1..1....x+1...2x...1...FN
A210600 u x....x+1..1....x+1...x+1..1...BF*
A210601 v x....x+1..1....x+1...x+1..1...BF*
A210597 u 2x...1....1....x+1...1....1...BF
A210601 v 2x...1....1....x+1...1....1...BFN*
A210603 u 2x...1....1....x+1...x+1..1...BF
A210738 v 2x...1....1....x+1...x+1..1...CBF*
A210739 u 2x...x....1....x+1...x....1...CF^
A210740 v 2x...x....1....x+1...x....1...DF*^
A210741 u 2x...x....1....x+1...x+1..1...BCFO
A210742 v 2x...x....1....x+1...x+1..1...CFO*
A210743 u 2x...x+1..1....x+1...1....1...F
A210744 v 2x...x+1..1....x+1...1....1...FN
A210747 u 2x...x+1..1....x+1...x+1..1...FF
A210748 v 2x...x+1..1....x+1...x+1..1...CFF*
A210749 u x+1..1....1....x+1...2x...1...BCF
A210750 v x+1..1....1....x+1...2x...1...BF
A210751 u x+1..x....1....x+1...2x...1...FNT
A210752 v x+1..x....1....x+1...2x...1...FN
A210753 u x+1..x....1....x+1...x+1..1...BNZ*
A210754 v x+1..x....1....x+1...x+1..1...BCT*
A210755 u x+1..2x...1....x+1...x+1..1...N*
A210756 v x+1..2x...1....x+1...x+1..1...CT*
A210789 u 1....x....0....x+2...x-1..0...CFFN
A210790 v 1....x....0....x+2...x-1..0...CEFF
A210791 u 1....x....0....x-1...x+2..0...CFNP
A210792 v 1....x....0....x-1...x+2..0...CF
A210793 u 1....x+1..0....x+2...x-1..0...CFNP
A210794 v 1....x+1..0....x+2...x-1..0...FPP
A210795 u 1....x....1....x+2...x-1..0...FN
A210796 v 1....x....1....x+2...x-1..0...FO
A210797 u 1....x....0....x+2...x-1..1...CF
A210798 v 1....x....0....x+2...x-1..1...F
A210799 u 1....x+1..1....x+2...x-1..0...FN
A210800 v 1....x+1..1....x+2...x-1..0...F
A210801 u 1....x+1..1....x+2...x-1..1...FN
A210802 v 1....x+1..1....x+2...x-1..1...F
A210803 u 1....x....0....x-1...x+3..0...F*
A210804 v 1....x....0....x-1...x+3..0...F*
A210805 u 1....x....0....x+2...x-1.-1...CFFN
A210806 v 1....x....0....x+2...x-1.-1...FF
A210858 u 1....x....0....x+n...x....0...CFT*
A210859 v 1....x....0....x+n...x....0...FN*
A210860 u 1....x+1..0....x+n...x....0...F
A210861 v 1....x+1..0....x+n...x....0...F*
A210862 u 1....x....1....x+n-1.x....0...FN
A210863 v 1....x....1....x+n-1.x....0...FS
A210864 u 1....x....1....x+n...x....0...FN
A210865 v 1....x....1....x+n...x....0...FT
A210866 u 1....x....0....x+n...x...-x...CFT
A210867 v 1....x....0....x+n...x...-x...FN
A210868 u 1....x....0....x+1...x-1..0...BCFN
A210869 v 1....x....0....x+1...x-1..0...BBCFNZ
A210870 u 1....x....0....x+1...x-1..1...CFFN
A210871 v 1....x....0....x+1...x-1..1...CFF
A210872 u x....1...-1....x.....x....1...BDFZ^
A210873 v x....1...-1....x.....x....1...BCFN^
A210876 u x....1....1....x.....x....x...BCCF^
A210877 v x....1....1....x.....x....x...BDFNZ^
A210878 u x....2x...0....x+1...x....1...DFZ^
A210879 v x....2x...0....x+1...x....1...FC*^
Some of these triangles have irregular row lengths, making it difficult to retrieve individual rows/columns/diagonals without actually computing the recurrence. - Georg Fischer, Sep 04 2021

			First five rows:
1
1...1
1...3...1
1...5...4...1
1...7...9...5...1
First five polynomials u(n,x):
1
1 + x
1 + 3x + x^2
1 + 5x + 4x^2 + x^3
1 + 7x + 9x^2 + 5x^3 + x^4

Cf. A029653, A208508, A208509.

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A208510 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A029653 *)

from sympy import Poly
from sympy.abc import x
def u(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x)
def v(n, x): return 1 if n==1 else u(n - 1, x) + x*v(n - 1, x) + 1
def a(n): return Poly(u(n, x), x).all_coeffs()[::-1]
for n in range(1, 13): print(a(n)) # Indranil Ghosh, May 27 2017

u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=(x+1)*u(n-1,x)+x for n>2, with u(n,2)=x+1.

Corrected by Philippe Deléham, Apr 10 2012
Corrections and additions by Clark Kimberling, May 09 2012
Corrections in the overview by Georg Fischer, Sep 04 2021

A002943 a(n) = 2n(2*n+1).

Original entry on oeis.org

0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8190, 8556, 8930

Offset: 0

N. J. A. Sloane

a(n) is the number of edges in (n+1) X (n+1) square grid with all horizontal, vertical and diagonal segments filled in. - Asher Auel, Jan 12 2000
In other words, the edge count of the (n+1) X (n+1) king graph. - Eric W. Weisstein, Jun 20 2017
Write 0,1,2,... in clockwise spiral; sequence gives numbers on one of 4 diagonals. (See Example section.)
The identity (4*n+1)^2 - (4*n^2+2*n)*(2)^2 = 1 can be written as A016813(n)^2 - a(n)*2^2 = 1. - Vincenzo Librandi, Jul 20 2010 - Nov 25 2012
Starting with "6" = binomial transform of [6, 14, 8, 0, 0, 0, ...]. - Gary W. Adamson, Aug 27 2010
The hyper-Wiener index of the crown graph G(n) (n>=3). The crown graph G(n) is the graph with vertex set {x(1), x(2), ..., x(n), y(1), y(2), ..., y(n)} and edge set {(x(i), y(j)): 1 <= i,j <= n, i != j} (= the complete bipartite graph K(n,n) with horizontal edges removed). The Hosoya-Wiener polynomial of G(n) is n(n-1)(t+t^2)+nt^3. - Emeric Deutsch, Aug 29 2013
Sum of the numbers from n to 3n. - Wesley Ivan Hurt, Oct 27 2014

			64--65--66--67--68--69--70--71--72
|
63  36--37--38--39--40--41--42
|   |                       |
62  35  16--17--18--19--20  43
|   |   |               |   |
61  34  15   4---5---6  21  44
|   |   |    |       |  |   |
60  33  14   3   0   7  22  45
|   |   |    |   |   |  |   |
59  32  13   2---1   8  23  46
|   |   |            |  |   |
58  31  12--11--10---9  24  47
|   |                   |   |
57  30--29--28--27--26--25  48
|                           |
56--55--54--53--52--51--50--49

R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 2nd ed., 1994, p. 99.

T. D. Noe, Table of n, a(n) for n = 0..1000
Amelia Carolina Sparavigna, The groupoids of Mersenne, Fermat, Cullen, Woodall and other Numbers and their representations by means of integer sequences, Politecnico di Torino, Italy (2019), [math.NT].
Leo Tavares, Illustration: Twin Diamond Stars.
Eric Weisstein's World of Mathematics, Crown Graph.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, King Graph.
Eric Weisstein's World of Mathematics, Queen Graph.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A001477, A007395, A007494, A007742, A014105, A016813, A033954, A045896, A046092, A054000, A118729, A173511.
Same as A033951 except start at 0.
Sequences from spirals: A001107, A002939, A007742, A033951, A033952, A033953, A033954, A033989, A033990, A033991, this sequence, A033996, A033988.
Sequences on the four axes of the square spiral: Starting at 0: A001107, A033991, A007742, A033954; starting at 1: A054552, A054556, A054567, A033951.
Sequences on the four diagonals of the square spiral: Starting at 0: A002939 = 2*A000384, A016742 = 4*A000290, this sequence = 2*A014105, A033996 = 8*A000217; starting at 1: A054554, A053755, A054569, A016754.
Sequences obtained by reading alternate terms on the X and Y axes and the two main diagonals of the square spiral: Starting at 0: A035608, A156859, A002378 = 2*A000217, A137932 = 4*A002620; starting at 1: A317186, A267682, A002061, A080335.
Cf. A003154, A056220, A002939.

a002943 n = 2 * n * (2 * n + 1)  -- Reinhard Zumkeller, Jan 12 2014

[ 4*n^2+2*n: n in [0..50]]; // Vincenzo Librandi, Nov 25 2012

A002943 := proc(n)
    2*n*(2*n+1) ;
end proc: # R. J. Mathar, Jun 28 2013

LinearRecurrence[{3, -3, 1}, {0, 6, 20}, 40] (* Harvey P. Dale, Aug 11 2011 *)
Table[2 n (2 n + 1), {n, 0, 40}] (* Harvey P. Dale, Aug 11 2011 *)

a(n)=2*n*(2*n+1) \\ Charles R Greathouse IV, Nov 20 2012

a(n) = 4*n^2 + 2*n.
a(n) = 2*A014105(n). - Omar E. Pol, May 21 2008
a(n) = floor((2*n + 1/2)^2). - Reinhard Zumkeller, Feb 20 2010
a(n) = A007494(n) + A173511(n) = A007742(n) + n. - Reinhard Zumkeller, Feb 20 2010
a(n) = 8*n+a(n-1) - 2 with a(0)=0. - Vincenzo Librandi, Jul 20 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, Aug 11 2011
a(n+1) = A045896(2*n+1). - Reinhard Zumkeller, Dec 12 2011
G.f.: 2*x*(3+x)/(1-x)^3. - Colin Barker, Jan 14 2012
From R. J. Mathar, Jan 15 2013: (Start)
Sum_{n>=1} 1/a(n) = 1 - log(2).
Sum_{n>=1} 1/a(n)^2 = 2*log(2) + Pi^2/6 - 3. (End)
a(n) = A118729(8*n+5). - Philippe Deléham, Mar 26 2013
a(n) = 1*A001477(n) + 2*A000217(n) + 3*A000290(n). - J. M. Bergot, Apr 23 2014
a(n) = 2 * A000217(2*n) = 2 * A014105(n). - Jon Perry, Oct 27 2014
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/4 + log(2)/2 - 1. - Amiram Eldar, Feb 22 2022
a(n) = A003154(n+1) - A056220(n+1). - Leo Tavares, Mar 31 2022
E.g.f.: 2*exp(x)*x*(3 + 2*x). - Stefano Spezia, Apr 24 2024
a(n) = A002939(-n) for all n in Z. - Charles Kusniec, Aug 12 2025

Formula fixed by Reinhard Zumkeller, Apr 09 2010

cp's OEIS Frontend

A000012 The simplest sequence of positive numbers: the all 1's sequence.

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A002378 Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).

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A005843 The nonnegative even numbers: a(n) = 2n.

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A000004 The zero sequence.

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A208510 Triangle of coefficients of polynomials u(n,x) jointly generated with A029653; see the Formula section.

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A002943 a(n) = 2n(2*n+1).