A000601 -id:A000601 - cp's OEIS frontend

A002620 Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).

0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400, 420, 441, 462, 484, 506, 529, 552, 576, 600, 625, 650, 676, 702, 729, 756, 784, 812

Offset: 0

N. J. A. Sloane

b(n) = a(n+2) is the number of multigraphs with loops on 2 nodes with n edges [so g.f. for b(n) is 1/((1-x)^2*(1-x^2))]. Also number of 2-covers of an n-set; also number of 2 X n binary matrices with no zero columns up to row and column permutation. - Vladeta Jovovic, Jun 08 2000
a(n) is also the maximal number of edges that a triangle-free graph of n vertices can have. For n = 2m, the maximum is achieved by the bipartite graph K(m, m); for n = 2m + 1, the maximum is achieved by the bipartite graph K(m, m + 1). - Avi Peretz (njk(AT)netvision.net.il), Mar 18 2001
a(n) is the number of arithmetic progressions of 3 terms and any mean which can be extracted from the set of the first n natural numbers (starting from 1). - Santi Spadaro, Jul 13 2001
This is also the order dimension of the (strong) Bruhat order on the Coxeter group A_{n-1} (the symmetric group S_n). - Nathan Reading (reading(AT)math.umn.edu), Mar 07 2002
Let M_n denote the n X n matrix m(i,j) = 2 if i = j; m(i, j) = 1 if (i+j) is even; m(i, j) = 0 if i + j is odd, then a(n+2) = det M_n. - Benoit Cloitre, Jun 19 2002
Sums of pairs of neighboring terms are triangular numbers in increasing order. - Amarnath Murthy, Aug 19 2002
Also, from the starting position in standard chess, minimum number of captures by pawns of the same color to place n of them on the same file (column). Beyond a(6), the board and number of pieces available for capture are assumed to be extended enough to accomplish this task. - Rick L. Shepherd, Sep 17 2002
For example, a(2) = 1 and one capture can produce "doubled pawns", a(3) = 2 and two captures is sufficient to produce tripled pawns, etc. (Of course other, uncounted, non-capturing pawn moves are also necessary from the starting position in order to put three or more pawns on a given file.) - Rick L. Shepherd, Sep 17 2002
Terms are the geometric mean and arithmetic mean of their neighbors alternately. - Amarnath Murthy, Oct 17 2002
Maximum product of two integers whose sum is n. - Matthew Vandermast, Mar 04 2003
a(n+2) gives number of non-symmetric partitions of n into at most 3 parts, with zeros used as padding. E.g., a(7) = 12 because we can write 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. - Jon Perry, Jul 08 2003
a(n-1) gives number of distinct elements greater than 1 of non-symmetric partitions of n into at most 3 parts, with zeros used as padding, appear in the middle. E.g., 5 = 5 + 0 + 0 = 0 + 5 + 0 = 4 + 1 + 0 = 1 + 4 + 0 = 1 + 0 + 4 = 3 + 2 + 0 = 2 + 3 + 0 = 2 + 0 + 3 = 2 + 2 + 1 = 2 + 1 + 2 = 3 + 1 + 1 = 1 + 3 + 1. Of these, 050, 140, 320, 230, 221, 131 qualify and a(4) = 6. - Jon Perry, Jul 08 2003
Union of square numbers (A000290) and oblong numbers (A002378). - Lekraj Beedassy, Oct 02 2003
Conjectured size of the smallest critical set in a Latin square of order n (true for n <= 8). - Richard Bean, Jun 12 2003 and Nov 18 2003
a(n) gives number of maximal strokes on complete graph K_n, when edges on K_n can be assigned directions in any way. A "stroke" is a locally maximal directed path on a directed graph. Examples: n = 3, two strokes can exist, "x -> y -> z" and " x -> z", so a(3) = 2. n = 4, four maximal strokes exist, "u -> x -> z" and "u -> y" and "u -> z" and "x -> y -> z", so a(4) = 4. - Yasutoshi Kohmoto, Dec 20 2003
Number of symmetric Dyck paths of semilength n+1 and having three peaks. E.g., a(4) = 4 because we have U*DUUU*DDDU*D, UU*DUU*DDU*DD, UU*DDU*DUU*DD and UUU*DU*DU*DDD, where U = (1, 1), D = (1, -1) and * indicates a peak. - Emeric Deutsch, Jan 12 2004
Number of valid inequalities of the form j + k < n + 1, where j and k are positive integers, j <= k, n >= 0. - Rick L. Shepherd, Feb 27 2004
See A092186 for another application.
Also, the number of nonisomorphic transversal combinatorial geometries of rank 2. - Alexandr S. Radionov (rasmailru(AT)mail.ru), Jun 02 2004
a(n+1) is the transform of n under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005
1, 2, 4, 6, 9, 12, 16, 20, 25, 30, ... specifies the largest number of copies of any of the gifts you receive on the n-th day in the "Twelve Days of Christmas" song. For example, on the fifth day of Christmas, you have 9 French hens. - Alonso del Arte, Jun 17 2005
a(n+1) is the number of noncongruent integer-sided triangles with largest side n. - David W. Wilson [Comment corrected Sep 26 2006]
A quarter-square table can be used to multiply integers since n*m = a(n+m) - a(n-m) for all integer n, m. - Michael Somos, Oct 29 2006
The sequence is the size of the smallest strong critical set in a Latin square of order n. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
Maximal number of squares (maximal area) in a polyomino with perimeter 2n. - Tanya Khovanova, Jul 04 2007
For n >= 3 a(n-1) is the number of bracelets with n+3 beads, 2 of which are red, 1 of which is blue. - Washington Bomfim, Jul 26 2008
Equals row sums of triangle A122196. - Gary W. Adamson, Nov 29 2008
Also a(n) is the number of different patterns of a 2-colored 3-partition of n. - Ctibor O. Zizka, Nov 19 2014
Also a(n-1) = C(((n+(n mod 2))/2), 2) + C(((n-(n mod 2))/2), 2), so this is the second diagonal of A061857 and A061866, and each even-indexed term is the average of its two neighbors. - Antti Karttunen
Equals triangle A171608 * ( 1, 2, 3, ...). - Gary W. Adamson, Dec 12 2009
a(n) gives the number of nonisomorphic faithful representations of the Symmetric group S_3 of dimension n. Any faithful representation of S_3 must contain at least one copy of the 2-dimensional irrep, along with any combination of the two 1-dimensional irreps. - Andrew Rupinski, Jan 20 2011
a(n+2) gives the number of ways to make change for "c" cents, letting n = floor(c/5) to account for the 5-repetitive nature of the task, using only pennies, nickels and dimes (see A187243). - Adam Sasson, Mar 07 2011
a(n) belongs to the sequence if and only if a(n) = floor(sqrt(a(n))) * ceiling(sqrt(a(n))), that is, a(n) = k^2 or a(n) = k*(k+1), k >= 0. - Daniel Forgues, Apr 17 2011
a(n) is the sum of the positive integers < n that have the opposite parity as n.
Deleting the first 0 from the sequence results in a sequence b = 0, 1, 2, 4, ... such that b(n) is sum of the positive integers <= n that have the same parity as n. The sequence b(n) is the additive counterpart of the double factorial. - Peter Luschny, Jul 06 2011
Third outer diagonal of Losanitsch's Triangle, A034851. - Fred Daniel Kline, Sep 10 2011
Written as a(1) = 1, a(n) = a(n-1) + ceiling (a(n-1)) this is to ceiling as A002984 is to floor, and as A033638 is to round. - Jonathan Vos Post, Oct 08 2011
a(n-2) gives the number of distinct graphs with n vertices and n regions. - Erik Hasse, Oct 18 2011
Construct the n-th row of Pascal's triangle (A007318) from the preceding row, starting with row 0 = 1. a(n) counts the total number of additions required to compute the triangle in this way up to row n, with the restrictions that copying a term does not count as an addition, and that all additions not required by the symmetry of Pascal's triangle are replaced by copying terms. - Douglas Latimer, Mar 05 2012
a(n) is the sum of the positive differences of the parts in the partitions of n+1 into exactly 2 parts. - Wesley Ivan Hurt, Jan 27 2013
a(n) is the maximum number of covering relations possible in an n-element graded poset. For n = 2m, this bound is achieved for the poset with two sets of m elements, with each point in the "upper" set covering each point in the "lower" set. For n = 2m+1, this bound is achieved by the poset with m nodes in an upper set covering each of m+1 nodes in a lower set. - Ben Branman, Mar 26 2013
a(n+2) is the number of (integer) partitions of n into 2 sorts of 1's and 1 sort of 2's. - Joerg Arndt, May 17 2013
Alternative statement of Oppermann's conjecture: For n>2, there is at least one prime between a(n) and a(n+1). - Ivan N. Ianakiev, May 23 2013. [This conjecture was mentioned in A220492, A222030. - Omar E. Pol, Oct 25 2013]
For any given prime number, p, there are an infinite number of a(n) divisible by p, with those a(n) occurring in evenly spaced clusters of three as a(n), a(n+1), a(n+2) for a given p. The divisibility of all a(n) by p and the result are given by the following equations, where m >= 1 is the cluster number for that p: a(2m*p)/p = p*m^2 - m; a(2m*p + 1)/p = p*m^2; a(2m*p + 2)/p = p*m^2 + m. The number of a(n) instances between clusters is 2*p - 3. - Richard R. Forberg, Jun 09 2013
Apart from the initial term this is the elliptic troublemaker sequence R_n(1,2) in the notation of Stange (see Table 1, p.16). For other elliptic troublemaker sequences R_n(a,b) see the cross references below. - Peter Bala, Aug 08 2013
a(n) is also the total number of twin hearts patterns (6c4c) packing into (n+1) X (n+1) coins, the coins left is A042948 and the voids left is A000982. See illustration in links. - Kival Ngaokrajang, Oct 24 2013
Partitions of 2n into parts of size 1, 2 or 4 where the largest part is 4, i.e., A073463(n,2). - Henry Bottomley, Oct 28 2013
a(n+1) is the minimum length of a sequence (of not necessarily distinct terms) that guarantees the existence of a (not necessarily consecutive) subsequence of length n in which like terms appear consecutively. This is also the minimum cardinality of an ordered set S that ensures that, given any partition of S, there will be a subset T of S so that the induced subpartition on T avoids the pattern ac/b, where a < b < c. - Eric Gottlieb, Mar 05 2014
Also the number of elements of the list 1..n+1 such that for any two elements {x,y} the integer (x+y)/2 lies in the range ]x,y[. - Robert G. Wilson v, May 22 2014
Number of lattice points (x,y) inside the region of the coordinate plane bounded by x <= n, 0 < y <= x/2. For a(11)=30 there are exactly 30 lattice points in the region below:
6|                                 .
.|                              .  |
5|                           .+__+
.|                        .  |  |  |
4|                     .+__++__+
.|                  .  |  |  |  |  |
3|               .+__++__++__+
.|            .  |  |  |  |  |  |  |
2|         .+__++__++__++__+
.|      .  |  |  |  |  |  |  |  |  |
1|   .+__++__++__++__++__+
.|.  |  |  |  |  |  |  |  |  |  |  |
0|.+__++__++__++__++__++_________
  0  1  2  3  4  5  6  7  8  9 10 11 .. n
  0  0  1  2  4  6  9 12 16 20 25 30 .. a(n) - Wesley Ivan Hurt, Oct 26 2014
a(n+1) is the greatest integer k for which there exists an n x n matrix M of nonnegative integers with every row and column summing to k, such that there do not exist n entries of M, all greater than 1, and no two of these entries in the same row or column. - Richard Stanley, Nov 19 2014
In a tiling of the triangular shape T_N with row length k for row k = 1, 2, ..., N >= 1 (or, alternatively row length N = 1-k for row k) with rectangular tiles, there can appear rectangles (i, j), N >= i >= j >= 1, of a(N+1) types (and their transposed shapes obtained by interchanging i and j). See the Feb 27 2004 comment above from Rick L. Shepherd. The motivation to look into this came from a proposal of Kival Ngaokrajang in A247139. - Wolfdieter Lang, Dec 09 2014
Every positive integer is a sum of at most four distinct quarter-squares; see A257018. - Clark Kimberling, Apr 15 2015
a(n+1) gives the maximal number of distinct elements of an n X n matrix which is symmetric (w.r.t. the main diagonal) and symmetric w.r.t. the main antidiagonal. Such matrices are called bisymmetric. See the Wikipedia link. - Wolfdieter Lang, Jul 07 2015
For 2^a(n+1), n >= 1, the number of binary bisymmetric n X n matrices, see A060656(n+1) and the comment and link by Dennis P. Walsh. - Wolfdieter Lang, Aug 16 2015
a(n) is the number of partitions of 2n+1 of length three with exactly two even entries (see below example). - John M. Campbell, Jan 29 2016
a(n) is the sum of the asymmetry degrees of all 01-avoiding binary words of length n. The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. a(6) = 9 because the 01-avoiding binary words of length 6 are 000000, 100000, 110000, 111000, 111100, 111110, and 111111, and the sum of their asymmetry degrees is 0 + 1 + 2 + 3 + 2 + 1 + 0 = 9. Equivalently, a(n) = Sum_{k>=0} k*A275437(n,k). - Emeric Deutsch, Aug 15 2016
a(n) is the number of ways to represent all the integers in the interval [3,n+1] as the sum of two distinct natural numbers. E.g., a(7)=12 as there are 12 different ways to represent all the numbers in the interval [3,8] as the sum of two distinct parts: 1+2=3, 1+3=4, 1+4=5, 1+5=6, 1+6=7, 1+7=8, 2+3=5, 2+4=6, 2+5=7, 2+6=8, 3+4=7, 3+5=8. - Anton Zakharov, Aug 24 2016
a(n+2) is the number of conjugacy classes of involutions (considering the identity as an involution) in the hyperoctahedral group C_2 wreath S_n. - Mark Wildon, Apr 22 2017
a(n+2) is the maximum number of pieces of a pizza that can be made with n cuts that are parallel or perpendicular to each other. - Anton Zakharov, May 11 2017
Also the matching number of the n X n black bishop graph. - Eric W. Weisstein, Jun 26 2017
The answer to a question posed by W. Mantel: a(n) is the maximum number of edges in an n-vertex triangle-free graph. Also solved by H. Gouwentak, J. Teixeira de Mattes, F. Schuh and W. A. Wythoff. - Charles R Greathouse IV, Feb 01 2018
Number of nonisomorphic outer planar graphs of order n >= 3, size n+2, and maximum degree 4. - Christian Barrientos and Sarah Minion, Feb 27 2018
Maximum area of a rectangle with perimeter 2n and sides of integer length. - André Engels, Jul 29 2018
Also the crossing number of the complete bipartite graph K_{3,n+1}. - Eric W. Weisstein, Sep 11 2018
a(n+2) is the number of distinct genotype frequency vectors possible for a sample of n diploid individuals at a biallelic genetic locus with a specified major allele. Such vectors are the lists of nonnegative genotype frequencies (n_AA, n_AB, n_BB) with n_AA + n_AB + n_BB = n and n_AA >= n_BB. - Noah A Rosenberg, Feb 05 2019
a(n+2) is the number of distinct real spectra (eigenvalues repeated according to their multiplicity) for an orthogonal n X n matrix. The case of an empty spectrum list is logically counted as one of those possibilities, when it exists. Thus a(n+2) is the number of distinct reduced forms (on the real field, in orthonormal basis) for elements in O(n). - Christian Devanz, Feb 13 2019
a(n) is the number of non-isomorphic asymmetric graphs that can be created by adding a single edge to a path on n+4 vertices. - Emma Farnsworth, Natalie Gomez, Herlandt Lino, and Darren Narayan, Jul 03 2019
a(n+1) is the number of integer triangles with largest side n. - James East, Oct 30 2019
a(n) is the number of nonempty subsets of {1,2,...,n} that contain exactly one odd and one even number.  For example, for n=7, a(7)=12 and the 12 subsets are {1,2}, {1,4}, {1,6}, {2,3}, {2,5}, {2,7}, {3,4}, {3,6}, {4,5}, {4,7}, {5,6}, {6,7}. - Enrique Navarrete, Dec 16 2019
a(n+1) is also the n-th term of the Saind sequence (w_n){n>=1}, i.e., the infinite sequence caused by the entries of the queue of the degree sequences associated with the Saind arrays, as n increases. - _Giulia Palma, Jun 24 2020
Aside from the first two terms, a(n) enumerates the number of distinct normal ordered terms in the expansion of the differential operator (x + d/dx)^m associated to the Hermite polynomials and the Heisenberg-Weyl algebra. It also enumerates the number of distinct monomials in the bivariate polynomials corresponding to the partial sums of the series for cos(x+y) and sin(x+y). Cf. A344678. - Tom Copeland, May 27 2021
a(n) is the maximal number of negative products a_i * a_j (1 <= i <= j <= n), where all a_i are real numbers. - Logan Pipes, Jul 08 2021
From Allan Bickle, Dec 20 2021: (Start)
a(n) is the maximum product of the chromatic numbers of a graph of order n-1 and its complement.  The extremal graphs are characterized in the papers of Finck (1968) and Bickle (2023).
a(n) is the maximum product of the degeneracies of a graph of order n+1 and its complement.  The extremal graphs are characterized in the paper of Bickle (2012). (End)
a(n) is the maximum number m such that m white rooks and m black rooks can coexist on an n-1 X n-1 chessboard without attacking each other. - Aaron Khan, Jul 13 2022
Partial sums of A004526. - Bernard Schott, Jan 06 2023
a(n) is the number of 231-avoiding odd Grassmannian permutations of size n. - Juan B. Gil, Mar 10 2023
a(n) is the number of integer tuples (x,y) satisfying n + x + y >= 0, 25*n + x - 11*y >=0, 25*n - 11*x + y >=0, n + x + y == 0 (mod 12) , 25*n + x - 11*y == 0 (mod 5), 25*n - 11*x + y == 0 (mod 5) . For n=2, the sole solution is (x,y) = (0,0) and so a(2) = 1. For n = 3, the a(3) = 2 solutions are (-3, 2) and (2, -3). - Jeffery Opoku, Feb 16 2024
Let us consider triangles whose vertices are the centers of three squares constructed on the sides of a right triangle. a(n) is the integer part of the area of these triangles, taken without repetitions and in ascending order. See the illustration in the links. - Nicolay Avilov, Aug 05 2024
For n>=2, a(n) is the indendence number of the 2-token graph F_2(P_n) of the path graph P_n on n vertices. (Alternatively, as noted by Peter Munn, F_2(P_n) is the nXn square lattice, or grid, graph diminished by a cut across the diagonal.) - Miquel A. Fiol, Oct 05 2024
For n >= 1, also the lower matching number of the n-triangular honeycomb rook graph. - Eric W. Weisstein, Dec 14 2024
a(n-1) is also the minimal number of edges that a graph of n vertices must have such that any 3 vertices share at least one edge. - Ruediger Jehn, May 20 2025
a(n) is the number of edges of the antiregular graph A_n.  This is the unique connected graph with n vertices and degrees 1 to n-1 (floor(n/2) repeated). - Allan Bickle, Jun 15 2025

			a(3) = 2, floor(3/2)*ceiling(3/2) = 2.
[ n] a(n)
---------
[ 2] 1
[ 3] 2
[ 4] 1 + 3
[ 5] 2 + 4
[ 6] 1 + 3 + 5
[ 7] 2 + 4 + 6
[ 8] 1 + 3 + 5 + 7
[ 9] 2 + 4 + 6 + 8
From _Wolfdieter Lang_, Dec 09 2014: (Start)
Tiling of a triangular shape T_N, N >= 1 with rectangles:
N=5, n=6: a(6) = 9 because all the rectangles (i, j) (modulo transposition, i.e., interchange of i and j) which are of use are:
  (5, 1)                ;  (1, 1)
  (4, 2), (4, 1)        ;  (2, 2), (2, 1)
                        ;  (3, 3), (3, 2), (3, 1)
That is (1+1) + (2+2) + 3 = 9 = a(6). Partial sums of 1, 1, 2, 2, 3, ... (A004526). (End)
Bisymmetric matrices B: 2 X 2, a(3) = 2 from B[1,1] and B[1,2]. 3 X 3, a(4) = 4 from B[1,1], B[1,2], B[1,3], and B[2,2]. - _Wolfdieter Lang_, Jul 07 2015
From _John M. Campbell_, Jan 29 2016: (Start)
Letting n=5, there are a(n)=a(5)=6 partitions of 2n+1=11 of length three with exactly two even entries:
(8,2,1) |- 2n+1
(7,2,2) |- 2n+1
(6,4,1) |- 2n+1
(6,3,2) |- 2n+1
(5,4,2) |- 2n+1
(4,4,3) |- 2n+1
(End)
From _Aaron Khan_, Jul 13 2022: (Start)
Examples of the sequence when used for rooks on a chessboard:
.
A solution illustrating a(5)=4:
  +---------+
  | B B . . |
  | B B . . |
  | . . W W |
  | . . W W |
  +---------+
.
A solution illustrating a(6)=6:
  +-----------+
  | B B . . . |
  | B B . . . |
  | B B . . . |
  | . . W W W |
  | . . W W W |
  +-----------+
(End)

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Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 36.
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T. Jenkyns and E. Muller, Triangular triples from ceilings to floors, Amer. Math. Monthly, Vol. 107 (Aug. 2000), pp. 634-639.
Paloma Jiménez-Sepúlveda and Luis Manuel Rivera, Independence numbers of some double vertex graphs and pair graphs, arXiv:1810.06354 [math.CO], 2018.
V. Jovovic, Vladeta Jovovic, Number of binary matrices.
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seq., Vol. 7 (2004), Article 04.1.6.
Jukka Kohonen, Counting graded lattices of rank three that have few coatoms, arXiv:1804.03679 [math.CO], 2018.
S. Lafortune, A. Ramani, B. Grammaticos, Y. Ohta and K.M. Tamizhmani, Blending two discrete integrability criteria: ..., arXiv:nlin/0104020 [nlin.SI], 2001.
W. Lanssens, B. Demoen and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber., Vol. 30 (1897), pp. 1917-1926.
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Index entries for two-way infinite sequences.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).
Index entries for "core" sequences.

A087811 is another version of this sequence.
Cf. A024206, A072280, A002984, A007590, A000212, A118015, A056827, A118013, A128174, A000601, A115514, A189151, A063657, A171608, A005044, A030179, A275437, A004526.
Differences of A002623. Complement of A049068.
a(n) = A014616(n-2) + 2 = A033638(n) - 1 = A078126(n) + 1. Cf. A055802, A055803.
Antidiagonal sums of array A003983.
Cf. A033436 - A033444. - Reinhard Zumkeller, Nov 30 2009
Cf. A008217, A014980, A197081, A197122.
Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A007590 (= R_n(2,4)), A030511 (= R_n(2,6) = R_n(4,6)), A033436 (= R_n(1,4) = R_n(3,4)), A033437 (= R_n(1,5) = R_n(4,5)), A033438 (= R_n(1,6) = R_n(5,6)), A033439 (= R_n(1,7) = R_n(6,7)), A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A077043, A060656 (2^a(n)), A344678.
Cf. A000290, A056834, A130519, A056838, A056865.
Cf. A250000 (queens on a chessboard), A176222 (kings on a chessboard), A355509 (knights on a chessboard).
Maximal product of k positive integers with sum n, for k = 2..10: this sequence (k=2), A006501 (k=3), A008233 (k=4), A008382 (k=5), A008881 (k=6), A009641 (k=7), A009694 (k=8), A009714 (k=9), A354600 (k=10).

# using the formula by Paul Barry
A002620 := List([1..10^4], n-> (2*n^2 - 1 + (-1)^n)/8); # Muniru A Asiru, Feb 01 2018

a002620 = (`div` 4) . (^ 2) -- Reinhard Zumkeller, Feb 24 2012

[ Floor(n/2)*Ceiling(n/2) : n in [0..40]];

A002620 := n->floor(n^2/4); G002620 := series(x^2/((1-x)^2*(1-x^2)),x,60);
with(combstruct):ZL:=[st,{st=Prod(left,right),left=Set(U,card=r),right=Set(U,card=1)}, unlabeled]: subs(r=1,stack): seq(count(subs(r=2,ZL),size=m),m=0..57) ; # Zerinvary Lajos, Mar 09 2007

Table[Ceiling[n/2] Floor[n/2], {n, 0, 56}] (* Robert G. Wilson v, Jun 18 2005 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 0, 1, 2}, 60] (* Harvey P. Dale, Oct 05 2012 *)
Table[Floor[n^2/4], {n, 0, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
Floor[Range[0, 20]^2/4] (* Eric W. Weisstein, Sep 11 2018 *)
CoefficientList[Series[-(x^2/((-1 + x)^3 (1 + x))), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 11 2018 *)
Table[Floor[n^2/2]/2, {n, 0, 56}] (* Clark Kimberling, Dec 05 2021 *)

makelist(floor(n^2/4),n,0,50); /* Martin Ettl, Oct 17 2012 */

PARI
```
a(n)=n^2\4
```

(t(n)=n*(n+1)/2);for(i=1,50,print1(",",(-1)^i*sum(k=1,i,(-1)^k*t(k))))

a(n)=n^2>>2 \\ Charles R Greathouse IV, Nov 11 2009

x='x+O('x^100); concat([0, 0], Vec(x^2/((1-x)^2*(1-x^2)))) \\ Altug Alkan, Oct 15 2015

def A002620(n): return (n**2)>>2 # Chai Wah Wu, Jul 07 2022

def A002620():
     x, y = 0, 1
     yield x
     while true:
         yield x
         x, y = x + y, x//y + 1
a = A002620(); print([next(a) for i in range(58)]) # Peter Luschny, Dec 17 2015

a(n) = (2*n^2-1+(-1)^n)/8. - Paul Barry, May 27 2003
G.f.: x^2/((1-x)^2*(1-x^2)) = x^2 / ( (1+x)*(1-x)^3 ). - Simon Plouffe in his 1992 dissertation, leading zeros dropped
E.g.f.: exp(x)*(2*x^2+2*x-1)/8 + exp(-x)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - Jaume Oliver Lafont, Dec 05 2008
a(-n) = a(n) for all n in Z.
a(n) = a(n-1) + floor(n/2), n > 0. Partial sums of A004526. - Adam Kertesz, Sep 20 2000
a(n) = a(n-1) + a(n-2) - a(n-3) + 1 [with a(-1) = a(0) = a(1) = 0], a(2k) = k^2, a(2k-1) = k(k-1). - Henry Bottomley, Mar 08 2000
0*0, 0*1, 1*1, 1*2, 2*2, 2*3, 3*3, 3*4, ... with an obvious pattern.
a(n) = Sum_{k=1..n} floor(k/2). - Yong Kong (ykong(AT)curagen.com), Mar 10 2001
a(n) = n*floor((n-1)/2) - floor((n-1)/2)*(floor((n-1)/2)+ 1); a(n) = a(n-2) + n-2 with a(1) = 0, a(2) = 0. - Santi Spadaro, Jul 13 2001
Also: a(n) = binomial(n, 2) - a(n-1) = A000217(n-1) - a(n-1) with a(0) = 0. - Labos Elemer, Apr 26 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k, 2). - Paul Barry, Jul 01 2003
a(n) = (-1)^n * partial sum of alternating triangular numbers. - Jon Perry, Dec 30 2003
a(n) = A024206(n+1) - n. - Philippe Deléham, Feb 27 2004
a(n) = a(n-2) + n - 1, n > 1. - Paul Barry, Jul 14 2004
a(n+1) = Sum_{i=0..n} min(i, n-i). - Marc LeBrun, Feb 15 2005
a(n+1) = Sum_{k = 0..floor((n-1)/2)} n-2k; a(n+1) = Sum_{k=0..n} k*(1-(-1)^(n+k-1))/2. - Paul Barry, Apr 16 2005
a(n) = A108561(n+1,n-2) for n > 2. - Reinhard Zumkeller, Jun 10 2005
1 + 1/(1 + 2/(1 + 4/(1 + 6/(1 + 9/(1 + 12/(1 + 16/(1 + ...))))))) = 6/(Pi^2 - 6) = 1.550546096730... - Philippe Deléham, Jun 20 2005
a(n) = Sum_{k=0..n} Min_{k, n-k}, sums of rows of the triangle in A004197. - Reinhard Zumkeller, Jul 27 2005
For n > 2 a(n) = a(n-1) + ceiling(sqrt(a(n-1))). - Jonathan Vos Post, Jan 19 2006
Sequence starting (2, 2, 4, 6, 9, ...) = A128174 (as an infinite lower triangular matrix) * vector [1, 2, 3, ...]; where A128174 = (1; 0,1; 1,0,1; 0,1,0,1; ...). - Gary W. Adamson, Jul 27 2007
a(n) = Sum_{i=k..n} P(i, k) where P(i, k) is the number of partitions of i into k parts. - Thomas Wieder, Sep 01 2007
a(n) = sum of row (n-2) of triangle A115514. - Gary W. Adamson, Oct 25 2007
For n > 1: gcd(a(n+1), a(n)) = a(n+1) - a(n). - Reinhard Zumkeller, Apr 06 2008
a(n+3) = a(n) + A000027(n) + A008619(n+1) = a(n) + A001651(n+1) with a(1) = 0, a(2) = 0, a(3) = 1. - Yosu Yurramendi, Aug 10 2008
a(2n) = A000290(n). a(2n+1) = A002378(n). - Gary W. Adamson, Nov 29 2008
a(n+1) = a(n) + A110654(n). - Reinhard Zumkeller, Aug 06 2009
a(n) = Sum_{k=0..n} (k mod 2)*(n-k); Cf. A000035, A001477. - Reinhard Zumkeller, Nov 05 2009
a(n-1) = (n*n - 2*n + n mod 2)/4. - Ctibor O. Zizka, Nov 23 2009
a(n) = round((2*n^2-1)/8) = round(n^2/4) = ceiling((n^2-1)/4). - Mircea Merca, Nov 29 2010
n*a(n+2) = 2*a(n+1) + (n+2)*a(n). Holonomic Ansatz with smallest order of recurrence. - Thotsaporn Thanatipanonda, Dec 12 2010
a(n+1) = (n*(2+n) + n mod 2)/4. - Fred Daniel Kline, Sep 11 2011
a(n) = A199332(n, floor((n+1)/2)). - Reinhard Zumkeller, Nov 23 2011
a(n) = floor(b(n)) with b(n) = b(n-1) + n/(1+e^(1/n)) and b(0)= 0. - Richard R. Forberg, Jun 08 2013
a(n) = Sum_{i=1..floor((n+1)/2)} (n+1)-2i. - Wesley Ivan Hurt, Jun 09 2013
a(n) = floor((n+2)/2 - 1)*(floor((n+2)/2)-1 + (n+2) mod 2). - Wesley Ivan Hurt, Jun 09 2013
Sum_{n>=2} 1/a(n) = 1 + zeta(2) = 1+A013661. - Enrique Pérez Herrero, Jun 30 2013
Empirical: a(n-1) = floor(n/(e^(4/n)-1)). - Richard R. Forberg, Jul 24 2013
a(n) = A007590(n)/2. - Wesley Ivan Hurt, Mar 08 2014
A237347(a(n)) = 3; A235711(n) = A003415(a(n)). - Reinhard Zumkeller, Mar 18 2014
A240025(a(n)) = 1. - Reinhard Zumkeller, Jul 05 2014
0 = a(n)*a(n+2) + a(n+1)*(-2*a(n+2) + a(n+3)) for all integers n. - Michael Somos, Nov 22 2014
a(n) = Sum_{j=1..n} Sum_{i=1..n} ceiling((i+j-n-1)/2). - Wesley Ivan Hurt, Mar 12 2015
a(4n+1) = A002943(n) for all n>=0. - M. F. Hasler, Oct 11 2015
a(n+2)-a(n-2) = A004275(n+1). - Anton Zakharov, May 11 2017
a(n) = floor(n/2)*floor((n+1)/2). - Bruno Berselli, Jun 08 2017
a(n) = a(n-3) + floor(3*n/2) - 2. - Yuchun Ji, Aug 14 2020
a(n)+a(n+1) = A000217(n). - R. J. Mathar, Mar 13 2021
a(n) = A004247(n,floor(n/2)). - Logan Pipes, Jul 08 2021
a(n) = floor(n^2/2)/2. - Clark Kimberling, Dec 05 2021
Sum_{n>=2} (-1)^n/a(n) = Pi^2/6 - 1. - Amiram Eldar, Mar 10 2022

A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

Original entry on oeis.org

0, 1, 11, 58, 177, 437, 894, 1659, 2813, 4502, 6836, 10008, 14121, 19449, 26117, 34372, 44422, 56597, 71044, 88160, 108115, 131328, 158074, 188773, 223604, 263172, 307719, 357715, 413493, 475690, 544480, 620632, 704381, 796413

Offset: 0

Clark Kimberling, Apr 27 2012

nonn

Each sequence in the following guide counts 4-tuples
(w,x,y,z) such that the indicated relation holds and the four numbers w,x,y,z are in {1,...,n}.  The notation "m div" means that m divides every term of the sequence.
A211058 ... wx <= yz
A211787 ... wx <= 2yz
A211795 ... wx < 2yz
A211797 ... wx > 2yz
A211809 ... wx >= 2yz
A211812 ... wx <= 3yz
A211917 ... wx < 3yz
A211918 ... wx > 3yz
A211919 ... wx >= 3yz
A211920 ... 2wx < 3yz
A211921 ... 2wx <= 3yz
A211922 ... 2wx > 3yz
A211923 ... 2wx >= 3yz
A212019 ... wx = 2yz ..... 2 div
A212020 ... wx = 3yz ..... 2 div
A212021 ... 2wx = 3yz .... 2 div
A212047 ... wx = 4yz
A212048 ... 3wx = 4yz .... 2 div
A212049 ... wx = 5yz ..... 2 div
A212050 ... 2wx = 5yz .... 2 div
A212051 ... 3wx = 5yz .... 2 div
A212052 ... 4wx = 5yz .... 2 div
A209978 ... wx = yz + 1 .. 2 div
A212053 ... wx <= yz + 1
A212054 ... wx > yz + 1
A212055 ... wx <= yz + 2
A212056 ... wx > yz + 2
A197168 ... wx = yz + 2 .. 2 div
A061201 ... w = xyz
A212057 ... w < xyz
A212058 ... w >= xyz
A212059 ... w = xyz - 1
A212060 ... w = xyz - 2
A212061 ... wx = (yz)^2
A212062 ... w^2 = xyz
A212063 ... w^2 < xyz
A212064 ... w^2 >= xyz
A212065 ... w^2 <= xyz
A212066 ... w^2 > xyz
A212067 ... w^3 = xyz
A002623 ... w = 2x + y + z
A006918 ... w = 2x + 2y + z
A000601 ... w = x + 2y + 3z (except for initial 0's)
A212068 ... 2w = x + y + z
A212069 ... 3w = x + y + z (w = average{x,y,z})
A212088 ... 3w < x + y + z
A212089 ... 3w >= x + y + z
A212090 ... w < x + y + z
A000332 ... w >= x + y + z
A212145 ... w < 2x + y + z
A001752 ... w >= 2x + y + z
A001400 ... w = 2x +3y + 4z
A005900 ... w = -x + y + z
A192023 ... w = -x + y + z + 2
A212091 ... w^2 = x^2 + y^2 + z^2 ... 3 div
A212087 ... w^2 + x^2 = y^2 + z^2
A212092 ... w^2 < x^2 + y^2 + z^2
A212093 ... w^2 <= x^2 + y^2 + z^2
A212094 ... w^2 > x^2 + y^2 + z^2
A212095 ... w^2 >= x^2 + y^2 + z^2
A212096 ... w^3 = x^3 + y^3 + z^3 ... 6 div
A212097 ... w^3 < x^3 + y^3 + z^3
A212098 ... w^3 <= x^3 + y^3 + z^3
A212099 ... w^3 > x^3 + y^3 + z^3
A212100 ... w^3 >= x^3 + y^3 + z^3
A212101 ... wx^2 = yz^2
A212102 ... 1/w = 1/x + 1/y + 1/z
A212103 ... 3/w = 1/x + 1/y + 1/z; w = h.m. of {x,y,z}
A212104 ... 3/w >= 1/x + 1/y + 1/z; w >= h.m.
A212105 ... 3/w < 1/x + 1/y + 1/z; w < h.m.
A212106 ... 3/w > 1/x + 1/y + 1/z; w > h.m.
A212107 ... 3/w <= 1/x + 1/y + 1/z; w <= h.m.
A212133 ... median(w,x,y,z) = mean(w,x,y,z)
A212134 ... median(w,x,y,z) <= mean(w,x,y,z)
A212135 ... median(w,x,y,z) > mean(w,x,y,z)
A212241 ... wx + yz > n
A212243 ... 2wx + yz = n
A212244 ... w = xyz - n
A212245 ... w = xyz - 2n
A212246 ... 2w = x + y + z - n
A212247 ... 3w = x + y + z + n
A212249 ... 3w < x + y + z + n
A212250 ... 3w >= x + y + z + n
A212251 ... 3w = x + y + z + n + 1
A212252 ... 3w = x + y + z + n + 2
A212254 ... w = x + 2y + 3z - n
A212255 ... w^2 = mean(x^2, y^2, z^2)
A212256 ... 4/w = 1/x + 1/y +1/z + 1/n
In the list above, if the relation in the second column is of the form "w rel ax + by + cz" then the sequence is linearly recurrent.  In the list below, the same is true for expressions involving more than one relation.
A000332 ... w < x <= y < z .... C(n,4)
A000914 ... w < x <= y < z .... Stirling 1st kind
A000914 ... w < x <= y >= z ... Stirling 1st kind
A050534 ... w < x < y >= z .... tritriangular
A001296 ... w <= x <= y >= z .. 4-dim pyramidal
A006322 ... x < x > y >= z
A002418 ... w < x >= y < z
A050534 ... w < x >=y >= z
A212415 ... w < x >= y <= z
A001296 ... w < x >= y <= z
A212246 ... w <= x > y <= z
A006322 ... w <= x >= y <= z
A212501 ... w > x < y >= z
A212503 ... w < 2x and y < 2z ..... A (note below)
A212504 ... w < 2x and y > 2z ..... A
A212505 ... w < 2x and y >= 2z .... A
A212506 ... w <= 2x and y <= 2z ... A
A212507 ... w < 2x and y <= 2z .... B
A212508 ... w < 2x and y < 3z ..... C
A212509 ... w < 2x and y <= 3z .... C
A212510 ... w < 2x and y > 3z ..... C
A212511 ... w < 2x and y >= 3z .... C
A212512 ... w <= 2x and y < 3z .... C
A212513 ... w <= 2x and y <= 3z ... C
A212514 ... w <= 2x and y > 3z .... C
A212515 ... w <= 2x and y >= 3z ... C
A212516 ... w > 2x and y < 3z ..... C
A212517 ... w > 2x and y <= 3z .... C
A212518 ... w > 2x and y > 3z ..... C
A212519 ... w > 2x and y >= 3z .... C
A212520 ... w >= 2x and y < 3z .... C
A212521 ... w >= 2x and y <= 3z ... C
A212522 ... w >= 2x and y > 3z .... C
A212523 ... w + x < y + z
A212560 ... w + x <= y + z
A212561 ... w + x = 2y + 2z
A212562 ... w + x < 2y + 2z ....... B
A212563 ... w + x <= 2y + 2z ...... B
A212564 ... w + x > 2y + 2z ....... B
A212565 ... w + x >= 2y + 2z ...... B
A212566 ... w + x = 3y + 3z
A212567 ... 2w + 2x = 3y + 3z
A212570 ... |w - x| = |x - y| + |y - z|
A212571 ... |w - x| < |x - y| + |y - z| ... B ... 4 div
A212572 ... |w - x| <= |x - y| + |y - z| .. B
A212573 ... |w - x| > |x - y| + |y - z| ... B ... 2 div
A212574 ... |w - x| >= |x - y| + |y - z| .. B
A212575 ... 2|w - x| = |x - y| + |y - z|
A212576 ... |w - x| = 2|x - y| + 2|y - z|
A212577 ... |w - x| = 2|x - y| - |y - z|
A212578 ... 2|w - x| = |x - y| - |y - z|
A212579 ... min{|w-x|,|w-y|} = min{|x-y|,|x-z|}
A212692 ... w = |x - y| + |y - z| ............... 2 div
A212568 ... w < |x - y| + |y - z| ............... 2 div
A212573 ... w <= |x - y| + |y - z| .............. 2 div
A212574 ... w > |x - y| + |y - z|
A212575 ... w >= |x - y| + |y - z|
A212676 ... w + x = |x - y| + |y - z| ......... H
A212677 ... w + y = |x - y| + |y - z|
A212678 ... w + x + y = |x - y| + |y - z|
A006918 ... w + x + y + z = |x - y| + |y - z| . H
A212679 ... |x - y| = |y - z| ................. H
A212680 ... |x - y| = |y - z| + 1 ..............H 2 div
A212681 ... |x - y| < |y - z| ................... 2 div
A212682 ... |x - y| >= |y - z|
A212683 ... |x - y| = w + |y - z| ............... 2 div
A212684 ... |x - y| = n - w + |y - z|
A212685 ... |w - x| = w + |y - z|
A186707 ... |w - x| < w + |y - z| ... (Note D)
A212714 ... |w - x| >= w + |y - z| .......... H . 2 div
A212686 ... 2*|w - x| = n + |y - z| ............. 4 div
A212687 ... 2*|w - x| < n + |y - z| ......... B
A212688 ... 2*|w - x| < n + |y - z| ......... B . 2 div
A212689 ... 2*|w - x| > n + |y - z| ......... B . 2 div
A212690 ... 2*|w - x| <= n + |y - z| ........ B
A212691 ... w + |x - y| = |x - z| + |y - z| . E . 2 div
...
In the above lists, all the terms of (w,x,y,z) are in {1,...,n}, but in the next lists they are all in {0,...,n}, and sequences are all linearly recurrent.
  R=range{w,x,y,z}=max{w,x,y,z}-min{w,x,y,z}.
A212740 ... max{w,x,y,z} < 2*min{w,x,y,z} .... A
A212741 ... max{w,x,y,z} >= 2*min{w,x,y,z} ... A
A212742 ... max{w,x,y,z} <= 2*min{w,x,y,z} ... A
A212743 ... max{w,x,y,z} > 2*min{w,x,y,z} .... A . 2 div
A212744 ... w=range (=max-min) ............... E
A212745 ... w=max{w,x,y,z} - 2*min{w,x,y,z}
A212746 ... R is in {w,x,y,z} ................ E
A212569 ... R is not in {w,x,y,z} ............ E
A212749 ... w=R or xA212750 ... w=R or x=R or yA212751 ... w=R or x=R or yA212752 ... wR ......... A
A212753 ... wR or z>R ......... D
A212754 ... wR or y>R or z>R ......... D
A002415 ... w = x + R ........................ D
A212755 ... |w - x| = R ...................... D
A212756 ... 2w = x + R
A212757 ... 2w = R
A212758 ... w = floor(R/2)
A002413 ... w = floor((x+y+z/2))
A212759 ... w, x, y are even
A212760 ... w is even and x = y + z .......... E
A212761 ... w is odd and x and y are even .... F . 2 div
A212762 ... w and x are odd y is even ........ F . 2 div
A212763 ... w, x, y are odd .................. F
A212764 ... w, x, y are even and z is odd .... F
A030179 ... w and x are even and y and z odd
A212765 ... w is even and x,y,z are odd ...... F
A212766 ... w is even and x is odd ........... A . 2 div
A212767 ... w and x are even and w+x=y+z ..... E
A212889 ... R is even ........................ A
A212890 ... R is odd ......................... A . 2 div
A212742 ... w-x, x-y, y-z are all even ....... A
A212892 ... w-x, x-y, y-z are all odd ........ A
A212893 ... w-x, x-y, y-z have same parity ... A
A005915 ... min{|w-x|, |x-y|, |y-z|} = 0
A212894 ... min{|w-x|, |x-y|, |y-z|} = 1
A212895 ... min{|w-x|, |x-y|, |y-z|} = 2
A179824 ... min{|w-x|, |x-y|, |y-z|} > 0
A212896 ... min{|w-x|, |x-y|, |y-z|} <= 1
A212897 ... min{|w-x|, |x-y|, |y-z|} > 1
A212898 ... min{|w-x|, |x-y|, |y-z|} <= 2
A212899 ... min{|w-x|, |x-y|, |y-z|} > 2
A212901 ... |w-x| = |x-y| = |y-z|
A212900 ... |w-x|, |x-y|, |y-z| are distinct . G
A212902 ... |w-x| < |x-y| < |y-z| ............ G
A212903 ... |w-x| <= |x-y| <= |y-z| .......... G
A212904 ... |w-x| + |x-y| + |y-z| = n ........ H
A212905 ... |w-x| + |x-y| + |y-z| = 2n ....... H
...
Note A: A212503-A212506 (and others) have these recurrence coefficients: 2,2,-6,0,6,-2,-2,1.
B: 3,-1,-5,5,1,-3,1
C: 0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1
D: 4,-5,0,5,-4,1
E: 1,3,-3,-3,3,1,-1
F: 1,4,-4,-6,6,4,-4,-1,1
G: 2,1,-3,-1,1,3,-1,-2,1
H: 2,1,-4,1,2,-1

		Examples
					a(2)=11 counts these (w,x,y,z): (1,1,1,1), (1,1,1,2), (1,1,2,1), (2,1,2,1), (2,1,1,2), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,2,2), (2,1,2,2), (2,2,2,2).
		
	
References
			A. Barvinok, Lattice Points and Lattice Polytopes, Chapter 7 in Handbook of Discrete and Computational Geometry, CRC Press, 1997, 133-152.
P. Gritzmann and J. M. Wills, Lattice Points, Chapter 3.2 in Handbook of Convex Geometry, vol. B, North-Holland, 1993, 765-797.

		
Links
			Bo Gyu Jeong, Table of n, a(n) for n = 0..200

		
CrossrefsCf. A000583 (n^4), A210000, A211809, A212959.
Programs
			
				Mathematica
				t = Compile[{{n, _Integer}}, Module[{s = 0},
    (Do[If[w*x < 2 y*z, s = s + 1], {w, 1, #},
      {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]] (* A211795 *)
(* Peter J. C. Moses, Apr 13 2012 *)
				

		
Formulaa(n) = n^4 - A211809(n).

A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120, 127, 133, 140, 147, 154, 161, 169, 176, 184, 192, 200, 208, 217, 225, 234, 243, 252, 261, 271, 280, 290, 300, 310, 320, 331, 341

Offset: 0

N. J. A. Sloane

Also number of tripods (trees with exactly 3 leaves) on n vertices. - Eric W. Weisstein, Mar 05 2011
Also number of partitions of n+3 into exactly 3 parts; number of partitions of n in which the greatest part is less than or equal to 3; and the number of nonnegative solutions to b + 2c + 3d = n.
Also a(n) gives number of partitions of n+6 into 3 distinct parts and number of partitions of 2n+9 into 3 distinct and odd parts, e.g., 15 = 11 + 3 + 1 = 9 + 5 + 1 = 7 + 5 + 3. - Jon Perry, Jan 07 2004
Also bracelets with n+3 beads 3 of which are red (so there are 2 possibilities with 5 beads).
More generally, the number of partitions of n into at most k parts is also the number of partitions of n+k into k positive parts, the number of partitions of n+k in which the greatest part is k, the number of partitions of n in which the greatest part is less than or equal to k, the number of partitions of n+k(k+1)/2 into exactly k distinct positive parts, the number of nonnegative solutions to b + 2c + 3d + ... + kz = n and the number of nonnegative solutions to 2c + 3d + ... + kz <= n. - Henry Bottomley, Apr 17 2001
Also coefficient of q^n in the expansion of (m choose 3)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
From Winston C. Yang (winston(AT)cs.wisc.edu), Apr 30 2002: (Start)
Write 1,2,3,4,... in a hexagonal spiral around 0, then a(n) for n > 0 is formed by the folding points (including the initial 1). The spiral begins:
.
                 85--84--83--82--81--80
                 /                     \
               86  56--55--54--53--52  79
               /   /                 \   \
             87  57  33--32--31--30  51  78
             /   /   /             \   \   \
           88  58  34  16--15--14  29  50  77
           /   /   /   /         \   \   \   \
         89  59  35  17   5---4  13  28  49  76
         /   /   /   /   /     \   \   \   \   \
       90  60  36  18   6   0   3  12  27  48  75
       /   /   /   /   /   /   /   /   /   /   /
     91  61  37  19   7   1---2  11  26  47  74
           \   \   \   \         /   /   /   /
           62  38  20   8---9--10  25  46  73
             \   \   \             /   /   /
             63  39  21--22--23--24  45  72
               \   \                 /   /
               64  40--41--42--43--44  71
                 \                     /
                 65--66--67--68--69--70
.
a(p) is maximal number of hexagons in a polyhex with perimeter at most 2p + 6. (End)
a(n-3) is the number of partitions of n into 3 distinct parts, where 0 is allowed as a part. E.g., at n=9, we can write 8+1+0, 7+2+0, 6+3+0, 4+5+0, 1+2+6, 1+3+5 and 2+3+4, which is a(6)=7. - Jon Perry, Jul 08 2003
a(n) gives number of partitions of n+6 into parts <=3 where each part is used at least once (subtract 6=1+2+3 from n). - Jon Perry, Jul 03 2004
This is also the number of partitions of n+3 into exactly 3 parts (there is a 1-to-1 correspondence between the number of partitions of n+3 in which the greatest part is 3 and the number of partitions of n+3 into exactly three parts). - Graeme McRae, Feb 07 2005
Apply the Riordan array (1/(1-x^3),x) to floor((n+2)/2). - Paul Barry, Apr 16 2005
Also, number of triangles that can be created with odd perimeter 3,5,7,9,11,... with all sides whole numbers. Note that triangles with even perimeter can be generated from the odd ones by increasing each side by 1. E.g., a(1) = 1 because perimeter 3 can make {1,1,1} 1 triangle. a(4) = 3 because perimeter 9 can make {1,4,4} {2,3,4} {3,3,3} 3 possible triangles. - Bruce Love (bruce_love(AT)ofs.edu.sg), Nov 20 2006
Also number of nonnegative solutions of the Diophantine equation x+2*y+3*z=n, cf. Pólya/Szegő reference.
From Vladimir Shevelev, Apr 23 2011: (Start)
Also a(n-3), n >= 3, is the number of non-equivalent necklaces of 3 beads each of them painted by one of n colors.
The sequence {a(n-3), n >= 3} solves the so-called Reis problem about convex k-gons in case k=3 (see our comment to A032279).
a(n-3) (n >= 3) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with three 1's in every row. (End)
A001399(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w = 2*x+3*y. - Clark Kimberling, Jun 04 2012
Also, for n >= 3, a(n-3) is the number of the distinct triangles in an n-gon, see the Ngaokrajang links. - Kival Ngaokrajang, Mar 16 2013
Also, a(n) is the total number of 5-curve coin patterns (5C4S type: 5 curves covering full 4 coins and symmetry) packing into fountain of coins base (n+3). See illustration in links. - Kival Ngaokrajang, Oct 16 2013
Also a(n) = half the number of minimal zero sequences for Z_n of length 3 [Ponomarenko]. - N. J. A. Sloane, Feb 25 2014
Also, a(n) equals the number of linearly-independent terms at 2n-th order in the power series expansion of an Octahedral Rotational Energy Surface (cf. Harter & Patterson). - Bradley Klee, Jul 31 2015
Also Molien series for invariants of finite Coxeter groups D_3 and A_3. - N. J. A. Sloane, Jan 10 2016
Number of different distributions of n+6 identical balls in 3 boxes as x,y,z where 0 < x < y < z. - Ece Uslu and Esin Becenen, Jan 11 2016
a(n) is also the number of partitions of 2*n with <= n parts and no part >= 4. The bijection to partitions of n with no part >= 4 is: 1 <-> 2, 2 <-> 1 + 3, 3 <-> 3 + 3 (observing the order of these rules). The <- direction uses the following fact for partitions of 2*n with <= n parts and no part >=4: for each part 1 there is a part 3, and an even number (including 0) of remaining parts 3. - Wolfdieter Lang, May 21 2019
List of the terms in A000567(n>=1), A049450(n>=1), A033428(n>=1), A049451(n>=1), A045944(n>=1), and A003215(n) in nondecreasing order. List of the numbers A056105(n)-1, A056106(n)-1, A056107(n)-1, A056108(n)-1, A056109(n)-1, and A003215(m) with n >= 1 and m >= 0 in nondecreasing order. Numbers of the forms 3n*(n-1)+1, n*(3n-2), n*(3n-1), 3n^2, n*(3n+1), n*(3n+2) with n >= 1 listed in nondecreasing order. Integers m such that lattice points from 1 through m on a hexagonal spiral starting at 1 forms a convex polygon. - Ya-Ping Lu, Jan 24 2024

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 7*x^6 + 8*x^7 + 10*x^8 + 12*x^9 + ...
Recall that in a necklace the adjacent beads have distinct colors. Suppose we have n colors with labels 1,...,n. Two colorings of the beads are equivalent if the cyclic sequences of the distances modulo n between labels of adjacent colors have the same period. If n=4, all colorings are equivalent. E.g., for the colorings {1,2,3} and {1,2,4} we have the same period {1,1,2} of distances modulo 4. So, a(n-3)=a(1)=1. If n=5, then we have two such periods {1,1,3} and {1,2,2} modulo 5. Thus a(2)=2. - _Vladimir Shevelev_, Apr 23 2011
a(0) = 1, i.e., {1,2,3} Number of different distributions of 6 identical balls to 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
a(3) = 3, i.e., {1,2,6}, {1,3,5}, {2,3,4} Number of different distributions of 9 identical balls in 3 boxes as x,y and z where 0 < x < y < z. - _Ece Uslu_, Esin Becenen, Jan 11 2016
From _Gus Wiseman_, Apr 15 2019: (Start)
The a(0) = 1 through a(8) = 10 integer partitions of n with at most three parts are the following. The Heinz numbers of these partitions are given by A037144.
  ()  (1)  (2)   (3)    (4)    (5)    (6)    (7)    (8)
           (11)  (21)   (22)   (32)   (33)   (43)   (44)
                 (111)  (31)   (41)   (42)   (52)   (53)
                        (211)  (221)  (51)   (61)   (62)
                               (311)  (222)  (322)  (71)
                                      (321)  (331)  (332)
                                      (411)  (421)  (422)
                                             (511)  (431)
                                                    (521)
                                                    (611)
The a(0) = 1 through a(7) = 8 integer partitions of n + 3 whose greatest part is 3 are the following. The Heinz numbers of these partitions are given by A080193.
  (3)  (31)  (32)   (33)    (322)    (332)     (333)      (3322)
             (311)  (321)   (331)    (3221)    (3222)     (3331)
                    (3111)  (3211)   (3311)    (3321)     (32221)
                            (31111)  (32111)   (32211)    (33211)
                                     (311111)  (33111)    (322111)
                                               (321111)   (331111)
                                               (3111111)  (3211111)
                                                          (31111111)
Non-isomorphic representatives of the a(0) = 1 through a(5) = 5 unlabeled multigraphs with 3 vertices and n edges are the following.
  {}  {12}  {12,12}  {12,12,12}  {12,12,12,12}  {12,12,12,12,12}
            {13,23}  {12,13,23}  {12,13,23,23}  {12,13,13,23,23}
                     {13,23,23}  {13,13,23,23}  {12,13,23,23,23}
                                 {13,23,23,23}  {13,13,23,23,23}
                                                {13,23,23,23,23}
The a(0) = 1 through a(8) = 10 strict integer partitions of n - 6 with three parts are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A007304.
  (321)  (421)  (431)  (432)  (532)  (542)  (543)  (643)   (653)
                (521)  (531)  (541)  (632)  (642)  (652)   (743)
                       (621)  (631)  (641)  (651)  (742)   (752)
                              (721)  (731)  (732)  (751)   (761)
                                     (821)  (741)  (832)   (842)
                                            (831)  (841)   (851)
                                            (921)  (931)   (932)
                                                   (A21)   (941)
                                                           (A31)
                                                           (B21)
The a(0) = 1 through a(8) = 10 integer partitions of n + 3 with three parts are the following. The Heinz numbers of these partitions are given by A014612.
  (111)  (211)  (221)  (222)  (322)  (332)  (333)  (433)  (443)
                (311)  (321)  (331)  (422)  (432)  (442)  (533)
                       (411)  (421)  (431)  (441)  (532)  (542)
                              (511)  (521)  (522)  (541)  (551)
                                     (611)  (531)  (622)  (632)
                                            (621)  (631)  (641)
                                            (711)  (721)  (722)
                                                   (811)  (731)
                                                          (821)
                                                          (911)
The a(0) = 1 through a(8) = 10 integer partitions of n whose greatest part is <= 3 are the following. The Heinz numbers of these partitions are given by A051037.
  ()  (1)  (2)   (3)    (22)    (32)     (33)      (322)      (332)
           (11)  (21)   (31)    (221)    (222)     (331)      (2222)
                 (111)  (211)   (311)    (321)     (2221)     (3221)
                        (1111)  (2111)   (2211)    (3211)     (3311)
                                (11111)  (3111)    (22111)    (22211)
                                         (21111)   (31111)    (32111)
                                         (111111)  (211111)   (221111)
                                                   (1111111)  (311111)
                                                              (2111111)
                                                              (11111111)
The a(0) = 1 through a(6) = 7 strict integer partitions of 2n+9 with 3 parts, all of which are odd, are the following. The Heinz numbers of these partitions are given by A307534.
  (5,3,1)  (7,3,1)  (7,5,1)  (7,5,3)   (9,5,3)   (9,7,3)   (9,7,5)
                    (9,3,1)  (9,5,1)   (9,7,1)   (11,5,3)  (11,7,3)
                             (11,3,1)  (11,5,1)  (11,7,1)  (11,9,1)
                                       (13,3,1)  (13,5,1)  (13,5,3)
                                                 (15,3,1)  (13,7,1)
                                                           (15,5,1)
                                                           (17,3,1)
The a(0) = 1 through a(8) = 10 strict integer partitions of n + 3 with 3 parts where 0 is allowed as a part (A = 10):
  (210)  (310)  (320)  (420)  (430)  (530)  (540)  (640)  (650)
                (410)  (510)  (520)  (620)  (630)  (730)  (740)
                       (321)  (610)  (710)  (720)  (820)  (830)
                              (421)  (431)  (810)  (910)  (920)
                                     (521)  (432)  (532)  (A10)
                                            (531)  (541)  (542)
                                            (621)  (631)  (632)
                                                   (721)  (641)
                                                          (731)
                                                          (821)
The a(0) = 1 through a(7) = 7 integer partitions of n + 6 whose distinct parts are 1, 2, and 3 are the following. The Heinz numbers of these partitions are given by A143207.
  (321)  (3211)  (3221)   (3321)    (32221)    (33221)     (33321)
                 (32111)  (32211)   (33211)    (322211)    (322221)
                          (321111)  (322111)   (332111)    (332211)
                                    (3211111)  (3221111)   (3222111)
                                               (32111111)  (3321111)
                                                           (32211111)
                                                           (321111111)
(End)
Partitions of 2*n with <= n parts and no part >= 4: a(3) = 3 from (2^3), (1,2,3), (3^2) mapping to (1^3), (1,2), (3), the partitions of 3 with no part >= 4, respectively. - _Wolfdieter Lang_, May 21 2019

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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.
Vadim Ponomarenko, Minimal zero sequences of finite cyclic groups, INTEGERS 4 (2004), #A24.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math. 35(5) (2004), 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Devansh Singh and S. N. Mishra, Representing K-parts Integer Partitions, Global Journal of Pure and Applied Mathematics (GJPAM), Volume 15 Number 6 (2019), pp. 889-907.
Karl Hermann Struve, Fresnel's Interferenzerscheinungen: Theoretisch und Experimentell Bearbeitet, Dorpat, 1881 (Thesis). [Gives the Round(n^2/12) formula.]
James Tanton, Young students approach integer triangles, FOCUS 22(5) (2002), 4 - 6.
James Tanton, Integer Triangles, Chapter 11 in "Mathematics Galore!" (MAA, 2012).
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See p. 11.
Richard Vale and Shayne Waldron, The construction of G-invariant finite tight frames, J. Four. Anal. Applic. 22 (2016), 1097-1120.
Eric Weisstein's World of Mathematics, Tripod.
Gus Wiseman, Number of ordered triples of distinct positive integers summing to n.
Gus Wiseman, Number of non-unimodal triples of distinct positive integers summing to n.
Gus Wiseman, Number of unimodal triples of distinct positive integers summing to n.
Winston C. Yang, Maximal and minimal polyhexes, 2002.
Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1,-1,1).
Index entries for Molien series.
Index entries for two-way infinite sequences

Cf. A008724, A003082, A117485, A026810, A026811, A026812, A026813, A026814, A026815, A026816, A000228, A036496, A008619, A001400, A001401, A069905, A008615, row 3 of A192517.
Molien series for finite Coxeter groups D_3 through D_12 are A001399, A051263, A266744-A266751.
Cf. A007304, A014612, A037144, A051037, A080193, A143207, A307534.

a001399 = p [1,2,3] where
   p _      0 = 1
   p []     _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Feb 28 2013

I:=[1,1,2,3,4,5]; [n le 6 select I[n] else Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5)+Self(n-6): n in [1..80]]; // Vincenzo Librandi, Feb 14 2015

[#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // Marius A. Burtea, Jan 06 2019

[Round((n+3)^2/12): n in [0..70]]; // Marius A. Burtea, Jan 06 2019

A001399 := proc(n)
    round( (n+3)^2/12) ;
end proc:
seq(A001399(n),n=0..40) ;
with(combstruct):ZL4:=[S,{S=Set(Cycle(Z,card<4))}, unlabeled]:seq(count(ZL4,size=n),n=0..61); # Zerinvary Lajos, Sep 24 2007
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=3)},unlabelled]: seq(combstruct[count](B, size=n), n=0..61); # Zerinvary Lajos, Mar 21 2009

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)), {x, 0, 65} ], x ]
Table[ Length[ IntegerPartitions[n, 3]], {n, 0, 61} ] (* corrected by Jean-François Alcover, Aug 08 2012 *)
k = 3; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
LinearRecurrence[{1,1,0,-1,-1,1},{1,1,2,3,4,5},70] (* Harvey P. Dale, Jun 21 2012 *)
a[ n_] := With[{m = Abs[n + 3] - 3}, Length[ IntegerPartitions[ m, 3]]]; (* Michael Somos, Dec 25 2014 *)
k=3 (* Number of red beads in bracelet problem *);CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)
Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&]],{n,0,30}] (* Gus Wiseman, Apr 15 2019 *)

{a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */

[round((n+3)**2 / 12) for n in range(0,62)] # Ya-Ping Lu, Jan 24 2024

G.f.: 1/((1 - x) * (1 - x^2) * (1 - x^3)) = -1/((x+1)*(x^2+x+1)*(x-1)^3); Simon Plouffe in his 1992 dissertation
a(n) = round((n + 3)^2/12). Note that this cannot be of the form (2*i + 1)/2, so ties never arise.
a(n) = A008284(n+3, 3), n >= 0.
a(n) = 1 + a(n-2) + a(n-3) - a(n-5) for all n in Z. - Michael Somos, Sep 04 2006
a(n) = a(-6 - n) for all n in Z. - Michael Somos, Sep 04 2006
a(6*n) = A003215(n), a(6*n + 1) = A000567(n + 1), a(6*n + 2) = A049450(n + 1), a(6*n + 3) = A033428(n + 1), a(6*n + 4) = A049451(n + 1), a(6*n + 5) = A045944(n + 1).
a(n) = a(n-1) + A008615(n+2) = a(n-2) + A008620(n) = a(n-3) + A008619(n) = A001840(n+1) - a(n-1) = A002620(n+2) - A001840(n) = A000601(n) - A000601(n-1). - Henry Bottomley, Apr 17 2001
P(n, 3) = (1/72) * (6*n^2 - 7 - 9*pcr{1, -1}(2, n) + 8*pcr{2, -1, -1}(3, n)) (see Comtet). [Here "pcr" stands for "prime circulator" and it is defined on p. 109 of Comtet, while the formula appears on p. 110. - Petros Hadjicostas, Oct 03 2019]
Let m > 0 and -3 <= p <= 2 be defined by n = 6*m+p-3; then for n > -3, a(n) = 3*m^2 + p*m, and for n = -3, a(n) = 3*m^2 + p*m + 1. - Floor van Lamoen, Jul 23 2001
72*a(n) = 17 + 6*(n+1)*(n+5) + 9*(-1)^n - 8*A061347(n). - Benoit Cloitre, Feb 09 2003
From Jon Perry, Jun 17 2003: (Start)
a(n) = 6*t(floor(n/6)) + (n%6) * (floor(n/6) + 1) + (n mod 6 == 0?1:0), where t(n) = n*(n+1)/2.
a(n) = ceiling(1/12*n^2 + 1/2*n) + (n mod 6 == 0?1:0).
[Here "n%6" means "n mod 6" while "(n mod 6 == 0?1:0)" means "if n mod 6 == 0 then 1, else 0" (as in C).]
(End)
a(n) = Sum_{i=0..floor(n/3)} 1 + floor((n - 3*i)/2). - Jon Perry, Jun 27 2003
a(n) = Sum_{k=0..n} floor((k + 2)/2) * (cos(2*Pi*(n - k)/3 + Pi/3)/3 + sqrt(3) * sin(2*Pi*(n-k)/3 + Pi/3)/3 + 1/3). - Paul Barry, Apr 16 2005
(m choose 3)_q = (q^m-1) * (q^(m-1) - 1) * (q^(m-2) - 1)/((q^3 - 1) * (q^2 - 1) * (q - 1)).
a(n) = Sum_{k=0..floor(n/2)} floor((3 + n - 2*k)/3). - Paul Barry, Nov 11 2003
A117220(n) = a(A003586(n)). - Reinhard Zumkeller, Mar 04 2006
a(n) = 3 * Sum_{i=2..n+1} floor(i/2) - floor(i/3). - Thomas Wieder, Feb 11 2007
Identical to the number of points inside or on the boundary of the integer grid of {I, J}, bounded by the three straight lines I = 0, I - J = 0 and I + 2J = n. - Jonathan Vos Post, Jul 03 2007
a(n) = A026820(n,3) for n > 2. - Reinhard Zumkeller, Jan 21 2010
Euler transform of length 3 sequence [ 1, 1, 1]. - Michael Somos, Feb 25 2012
a(n) = A005044(2*n + 3) = A005044(2*n + 6). - Michael Somos, Feb 25 2012
a(n) = A000212(n+3) - A002620(n+3). - Richard R. Forberg, Dec 08 2013
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6). - David Neil McGrath, Feb 14 2015
a(n) = floor((n^2+3)/12) + floor((n+2)/2). - Giacomo Guglieri, Apr 02 2019
From Devansh Singh, May 28 2020: (Start)
Let p(n, 3) be the number of 3-part integer partitions in which every part is > 0.
Then for n >= 3, p(n, 3) is equal to:
  (n^2 - 1)/12 when n is odd and 3 does not divide n.
  (n^2 + 3)/12 when n is odd and 3 divides n.
  (n^2 - 4)/12 when n is even and 3 does not divide n.
  (n^2)/12 when n is even and 3 divides n.
For n >= 3, p(n, 3) = a(n-3). (End)
a(n) = floor(((n+3)^2 + 4)/12). - Vladimír Modrák, Zuzana Soltysova, Dec 08 2020
Sum_{n>=0} 1/a(n) = 15/4 - Pi/(2*sqrt(3)) + Pi^2/18 + tanh(Pi/(2*sqrt(3)))*Pi/sqrt(3). - Amiram Eldar, Sep 29 2022
E.g.f.: exp(-x)*(9 + exp(2*x)*(47 + 42*x + 6*x^2) + 16*exp(x/2)*cos(sqrt(3)*x/2))/72. - Stefano Spezia, Mar 05 2023
a(6n) = 1+6*A000217(n); Sum_{i=1..n} a(6*i) = A000578(n+1). - David García Herrero, May 05 2024

Name edited by Gus Wiseman, Apr 15 2019

A002623 Expansion of 1/((1-x)^4*(1+x)).

Original entry on oeis.org

1, 3, 7, 13, 22, 34, 50, 70, 95, 125, 161, 203, 252, 308, 372, 444, 525, 615, 715, 825, 946, 1078, 1222, 1378, 1547, 1729, 1925, 2135, 2360, 2600, 2856, 3128, 3417, 3723, 4047, 4389, 4750, 5130, 5530, 5950, 6391, 6853, 7337, 7843, 8372, 8924, 9500

Offset: 0

N. J. A. Sloane

Also a(n) is the number of nondegenerate triangles that can be made from rods of lengths 1 to n+1. - Alfred Bruckstein; corrected by Hans Rudolf Widmer, Nov 02 2023
Also number of circumscribable (or escrible) quadrilaterals that can be made from rods of length 1,2,3,4,...,n. - Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr)
Also number of 2 X n binary matrices up to row and column permutation (see the link: Binary matrices up to row and column permutations). - Vladeta Jovovic
Also partial sum of alternate triangular numbers (1, 3, 1+6, 3+10, 1+6+15, 3+10+21, etc.); and also number of triangles pointing in opposite direction to largest triangle in triangular matchstick arrangement of side n+2 (cf. A002717, also the Larsen article). - Henry Bottomley, Aug 08 2000
Ordered union of A002412(n+1) and A016061(n+1). - Lekraj Beedassy, Oct 13 2003
Also Molien series for certain 4-D representation of cyclic group of order 2. - N. J. A. Sloane, Jun 12 2004
From Radu Grigore (radugrigore(AT)gmail.com), Jun 19 2004: (Start)
a(n) = floor( (n+2)*(n+4)*(2n+3) / 24 ). E.g., a(2) = floor(4*6*7/24) = 7 because there are 7 upside down triangles (6 of size one and 1 of size two) in the matchstick figure:
     /\
    /\/\
   /\/\/\
  /\/\/\/\
(End)
Number of non-congruent non-parallelogram trapezoids with positive integer sides (trapezints) and perimeter 2n+5. Also with perimeter 2n+8. - Michael Somos, May 12 2005
a(n) = A108561(n+4,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005
Also number of nonisomorphic planes with n points and 2 lines. E.g., a(0)=1 because with no points, we just have two empty lines. a(1)=3 because the one point may belong to 0, 1 or 2 lines. a(2)=7 because there are 7 ways to determine which of 2 points belong to which of 2 lines, up to isomorphism, i.e., up to a bijection f on the sets of points and a bijection g on the sets of lines, such that A belongs to a iff f(A) belongs to g(a). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
a(n-2) is the number of ways to pick two non-overlapping subwords of equal nonzero length from a word of length n. E.g., a(5-2)=a(3)=13 since the word 12345 of length 5 has the following subword pairs: 1,2; 1,3; 1,4; 1,5; 2,3; 2,4; 2,5; 3,4; 3,5; 4,5; 12,34; 12,45; 23,45. - Michael Somos, Oct 22 2006
Partial sums of A002620. - G.H.J. van Rees (vanrees(AT)cs.umanitoba.ca), Feb 16 2007
From Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007: (Start)
Also number of squares of any size in a staircase of n steps built with unit squares:
   
  ||__
  ||__|
  ||__||
For a staircase of 3 steps 6 squares of size 1 and 1 square of size 2, hence c(3)=7.
Columns sums of:
  1 3 6 10 15 21 28 ...
      1  3  6 10 15 ...
            1  3  6 ...
                  1 ...
  ---------------------
  1 3 7 13 22 34 50 ...
(End)
a(n) = sum of row n+1 of triangle A134446. Also, binomial transform of [1, 2, 2, 0, 1, -2, 4, -8, 16, -32, ...]. - Gary W. Adamson, Oct 25 2007
Let b(n) be the number of 4-tuples (w,x,y,z) having all terms in {1,...,n} and 2w=x+y+z+n; then b(n+3) = a(n) for n >= 0. - Clark Kimberling, May 08 2012
a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and w >= x+y and x <= y. - Clark Kimberling, Jun 04 2012
Also, number of unlabeled bipartite graphs with two left vertices and n right vertices. - Yavuz Oruc, Jan 14 2018
Also number of triples (x,y,z) with 0 < x <= y <= z <= n + 1, x + y > z. - Ralf Steiner, Feb 06 2020
Bisections A002412 and A016061: a(2*k) = k*(k+1)*(4*k-1)/3! and a(2*k+1) = (k+1)*(k+2)*(4*k+9)/3!, for k >= 0. See the Woolhouse link, II. Solution by Stephen Watson, p. 65, with index shifts. - Mo Li, Apr 02 2020
Also, Wiener index of the square of the path graph P_(n+2). - Allan Bickle, Aug 01 2020
Maximum Wiener index of all maximal 2-degenerate graphs with n+2 vertices.  (A maximal 2-degenerate graph can be constructed from a 2-clique by iteratively adding a new 2-leaf (vertex of degree 2) adjacent to two existing vertices.)  The extremal graphs are squares of paths, so the bound also applies to 2-trees and maximal outerplanar graphs. - Allan Bickle, Sep 15 2022

			G.f. = 1 + 3*x + 7*x^2 + 13*x^3 + 22*x^4 + 34*x^5 + 50*x^6 + 70*x^7 + 95*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 74, Problem 7.
P. Diaconis, R. L. Graham and B. Sturmfels, Primitive partition identities, in Combinatorics: Paul Erdős is Eighty, Vol. 2, Bolyai Soc. Math. Stud., 2, 1996, pp. 173-192.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts. Collection of articles dedicated to Professor P. L. Bhatnagar on his sixtieth birthday. Math. Student 40 (1972), 401-441 (1974).
I. Siap, Linear codes over F_2 + u*F_2 and their complete weight enumerators, in Codes and Designs (Ohio State, May 18, 2000), pp. 259-271. De Gruyter, 2002.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5, Lemma 6 5th line.
Allan Bickle and Zhongyuan Che, Wiener indices of maximal k-degenerate graphs, arXiv:1908.09202 [math.CO], 2019.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Shalosh B. Ekhad and Doron Zeilberger, Computerizing the Andrews-Fraenkel-Sellers Proofs on the Number of m-ary partitions mod m (and doing MUCH more!), arXiv preprint arXiv:1511.06791 [math.CO], 2015.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
Enrique González-Jiménez and Xavier Xarles, On a conjecture of Rudin on squares in Arithmetic Progressions, arXiv preprint arXiv:1301.5122 [math.NT], 2013.
H. Gupta, Partitions of j-partite numbers into twelve or a smaller number of parts, Math. Student 40 (1972), 401-441 (1974). [Annotated scanned copy]
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051. doi:10.1109/T-C.1973.223649.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 203
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 413
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]. See page 79.
W. Lanssens, B. Demoen, and P.-L. Nguyen, The Diagonal Latin Tableau and the Redundancy of its Disequalities, Report CW 666, July 2014, Department of Computer Science, KU Leuven
M. E. Larsen, The eternal triangle - a history of a counting problem, College Math. J., 20 (1989), 370-392.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
Math StackExchange, cycle index for S_2 X S_4, Apr. 2021
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218. See page 217.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Brian O'Sullivan and Thomas Busch, Spontaneous emission in ultra-cold spin-polarised anisotropic Fermi seas, arXiv 0810.0231v1 [quant-ph], 2008. [Eq 10a, lambda=2]
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
Giovanni Resta, Illustration for a(8)=70.
Marko Riedel, The O.g.f. of inequivalent colorings of a 2xN board with C colors in N is (1/2) 1/(1-w)^(C^2) + (1/2) 1/(1+w)^(C(C-1)/2)*1/(1-w)^(C(C+1)/2) by PET
Sebastian Sassi, Aula Al-Adulrazzaq, Matti Heikinheimo, and Kimmo Tuominen, Fast numerical evaluation of dark matter direct detection event rates, arXiv:2504.19714 [hep-ph], 2025. See p. 7.
Atsuto Seko, Tutorial: Systematic development of polynomial machine learning potentials for elemental and alloy systems, J. Appl. Phys. (2023) Vol. 133, 011101.
J. Silverman, V. E. Vickers and J. M. Mooney, On the number of Costas arrays as a function of array size, Proc. IEEE, 76 (1988), 851-853.
Eric Weisstein's World of Mathematics, Triangle Counting.
W. S. B. Woolhouse, Problem 2420. On the probability of the number of triangles, Mathematical questions with their solutions, v. 9 (June 1868), pp. 63-65.
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Cf. A002620 (first differences), A000292, A001752 (partial sums), A062109 (binomial transf.).
Bisections A002412, A016061.
Cf. also A002717 (a companion sequence), A002727, A006148, A057524, A134446, A014125, A122046, A122047.
Cf. A000217, A006918, A058187, A108561.
The maximum Wiener index of all maximal k-degenerate graphs for k=1..6 are given in A000292, A002623 (this sequence), A014125, A122046, A122047, A175724, respectively.

A002623 := n->(1/16)*(1+(-1)^n)+(n+1)/8+binomial(n+2,2)/4+binomial(n+3,3)/2;
seq( ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4,n=1..47); # Lewis
a := n -> ((-1)^n*3 + 45 + 68*n + 30*n^2 + 4*n^3) / 48:
seq(a(n), n=0..46); # Peter Luschny, Jan 22 2018

CoefficientList[Series[1/((1-x)^3(1-x^2)),{x,0,50}],x] (* or *) LinearRecurrence[{3,-2,-2,3,-1},{1,3,7,13,22},50] (* Harvey P. Dale, Jul 19 2011 *)
Table[((2 n^3 + 15 n^2 + 34 n + 45 / 2 + (3/2) (-1)^n) / 24), {n, 0, 100}] (* Vincenzo Librandi, Jan 15 2018 *)
a[ n_] := Floor[(n + 2)*(n + 4)*(2*n + 3)/24]; (* Michael Somos, Feb 19 2024 *)

{a(n) = (8 + 34/3*n + 5*n^2 + 2/3*n^3) \ 8}; /* Michael Somos, Sep 04 1999 */

x='x+O('x^50); Vec(1/((1 - x)^3 * (1 - x^2))) \\ Indranil Ghosh, Apr 04 2017

def A002623(n): return ((n+2)*(n+4)*((n<<1)+3)>>3)//3 # Chai Wah Wu, Mar 25 2024

a(n+1) = a(n) + {(k-1)*k if n=2*k} or {k*k if n=2*k+1}.
a(n)+a(n+1) = A000292(n+1).
a(n) = a(n-2) + A000217(n+1) = A002717(n+2) - A000292(n+1).
Also: a(n) = C(n+3, 3) - a(n-1) with a(0)=1. - Labos Elemer, Apr 26 2003
From Paul Barry, Jul 01 2003: (Start)
a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+3,3).
The signed version 1, -3, 7, ... has the formula:
a(n) = (4*n^3 + 30*n^2 + 68*n + 45)*(-1)^n/48 + 1/16.
This is the partial sums of the signed version of A000292. (End)
From Paul Barry, Jul 21 2003: (Start)
a(n) = Sum_{k=0..n} floor((k+2)^2/4).
a(n) = Sum_{k=0..n} Sum_{j=0..k} Sum_{i=0..j} (1+(-1)^i)/2. (End)
a(n) = a(n - 2) + (n*(n - 1))/2, with n>2, a(1)=0, a(2)=1; a(n) = (4*n^3+6*n^2-4*n+3*(-1)^n-3)/48, with offset 2. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004 (formula simplified by Bruno Berselli, Aug 29 2013)
a(n) = ((2*n+3)*(n+2)*(n+1)/6-floor((n+2)/2))/4, with offset 1. - Jerry W. Lewis (JLewis(AT)wyeth.com), Mar 23 2005
a(n) = 2*a(n-1) - a(n-2) + 1 + floor(n/2). - Bjorn Kjos-Hanssen (bjorn(AT)math.uconn.edu), Nov 10 2005
A002620(n+3) = a(n+1) - a(n). - Michael Somos, Sep 04 1999
Euler transform of length 2 sequence [ 3, 1]. - Michael Somos, Sep 04 2006
a(n) = -a(-5-n) for all n in Z. - Michael Somos, Sep 04 2006
Let P(i,k) be the number of integer partitions of n into k parts, then with k=2 we have a(n) = sum_{m=1}^{n} sum_{i=k}^{m} P(i,k). For k=1 we get A000217 = triangular numbers. - Thomas Wieder, Feb 18 2007
a(n) = (n+(3+(-1)^n)/2)*(n+(7+(-1)^n)/2)*(2*n+5-2*(-1)^n)/24. - Philippe LALLOUET (philip.lallouet(AT)orange.fr), Oct 19 2007 (corrected by Bruno Berselli, Aug 30 2013)
From Johannes W. Meijer, May 20 2011: (Start)
a(n) = A006918(n+1) + A006918(n).
a(n) = A058187(n-2) + 2*A058187(n-1) + A058187(n). (End)
a(0)=1, a(1)=3, a(2)=7, a(3)=13, a(4)=22; for n > 4, a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5). - Harvey P. Dale, Jul 19 2011
a(n) = Sum_{i=0..n+2} floor(i/2)*ceiling(i/2). - Bruno Berselli, Aug 30 2013
a(n) = 15/16 + (1/16)*(-1)^n + (17/12)*n + (5/8)*n^2 + (1/12)*n^3. - Robert Israel, Jul 07 2014
a(n) = Sum_{i=0..n+2} (n+1-i)*floor(i/2+1). - Bruno Berselli, Apr 04 2017
a(n) = 1 + floor((2*n^3 + 15*n^2 + 34*n) / 24). - Allan Bickle, Aug 01 2020
E.g.f.: ((24 + 51*x + 21*x^2 + 2*x^3)*cosh(x) + (21 + 51*x + 21*x^2 + 2*x^3)*sinh(x))/24. - Stefano Spezia, Jun 02 2021

A000602 Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 18, 35, 75, 159, 355, 802, 1858, 4347, 10359, 24894, 60523, 148284, 366319, 910726, 2278658, 5731580, 14490245, 36797588, 93839412, 240215803, 617105614, 1590507121, 4111846763, 10660307791, 27711253769

Offset: 0

N. J. A. Sloane

Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.
Ignoring stereoisomers means that the children of a node are unordered. They can be permuted in any way and it is still the same tree. See A000628 for the analogous sequence with stereoisomers counted.
In alkanes every carbon has valence exactly 4 and every hydrogen has valence exactly 1. But the trees considered here are just the carbon "skeletons" (with all the hydrogen atoms stripped off) so now each carbon bonds to 1 to 4 other carbons. The degree of each node is then <= 4.

			a(6)=5 because hexane has five isomers: n-hexane; 2-methylpentane; 3-methylpentane; 2,2-dimethylbutane; 2,3-dimethylbutane. - Michael Lugo (mtlugo(AT)mit.edu), Mar 15 2003 (corrected by _Andrey V. Kulsha_, Sep 22 2011)

Klemens Adam, Die Anzahlbestimmung der isomeren Alkane, MNU 1983, 36, 29 (in German).
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
L. Bytautats, D. J. Klein, Alkane Isomer Combinatorics: Stereostructure enumeration and graph-invariant and molecular-property distributions, J. Chem. Inf. Comput. Sci 39 (1999) 803, Table 1.
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Baeume genannt werden..., Chem. Ber. 8 (1875), 1056-1059.
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with More Realizable Isomers than the Observable Universe has Particles, Journal of Chemical Education, Vol. 66, 1989, pp. 278-281.
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.6.1 Chemical Isomers, p. 299.
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
Handbook of Combinatorics, North-Holland '95, p. 1963.
J. B. Hendrickson and C. A. Parks, "Generation and Enumeration of Carbon skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp. 101-107. See Table 2, column 2 on page 103.
M. D. Jackson and T. I. Bieber, Applications of degree distribution, 2: construction and enumeration of isomers in the alkane series, J. Chem. Info. and Computer Science, 33 (1993), 701-708.
J. Lederberg et al., Applications of artificial intelligence for chemical systems, I: The number of possible organic compounds. Acyclic structures containing C, H, O and N, J. Amer. Chem. Soc., 91 (1969), 2973-2097.
L. M. Masinter, Applications of artificial intelligence for chemical systems, XX, Exhaustive generation of cyclic and acyclic isomers, J. Amer. Chem. Soc., 96 (1974), 7702-7714.
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly - compare first link below]
M. Petkovsek and T. Pisanski, Counting disconnected structures: chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta, 78 (2005), 563-567.
D. H. Rouvray, An introduction to the chemical applications of graph theory, Congress. Numerant., 55 (1986), 267-280. - N. J. A. Sloane, Apr 08 2012
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).
Marten J. ten Hoor, Formula for Success?, Education in Chemistry, 2005, 42(1), 10.
S. Wagner, Graph-theoretical enumeration and digital expansions: an analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech. Physik, Tech. Univ. Graz, Austria, Feb., 2006.

Pontus von Brömssen, Table of n, a(n) for n = 0..1000 (terms n = 0..60 from N. J. A. Sloane)
Jean-François Alcover, Mathematica program translated from N. J. A. Sloane's Maple program for A000022, A000200, A000598, A000602, A000678
M. van Almsick, H. Dolhaine and H. Honig, Efficient algorithms to enumerate isomers and diamutamers with more than one type of substituent, J. Chem. Info. and Computer Science, 40 (2000), 956-966.
Vijayakumar Ambat, Article about OEIS in Malayalam that mentions this sequence, Mathrubhumi (daily newspaper in Kerala State), circa Nov 20 2022.
R. Aringhieri, P. Hansen and F. Malucelli, Chemical Tree Enumeration Algorithms, Report TR-99-09, Dept. Informatica, Univ. Pisa, 1999.
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes having n carbons and a longest chain of length d, J. Chem. Education, 65 (1988), 304-313.
H. Bottomley, Illustration of initial terms of A000022, A000200, A000602
L. Bytautas and D. J. Klein, Chemical combinatorics for alkane-isomer enumeration and more, J. Chem. Inf. Comput. Sci., 38 (1998), 1063-1078.
A. Cayley, Über die analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen, Chem. Ber. 8 (1875), 1056-1059. (Annotated scanned copy)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 478
Alfred W. Francis, Numbers of Isomeric Alkylbenzenes, J. Am. Chem. Soc., 69:6 (1947), pp. 1536-1537.
R. K. Guy, Letter to N. J. A. Sloane, Mar. 1988
F. Harary and R. Z. Norman, Dissimilarity characteristic theorems for graphs, Proc. Amer. Math. Soc. 11 (1960), 332-334.
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (8) (1931), 3077-3085.
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. (Annotated scanned copy)
H. R. Henze and C. M. Blair, The number of structurally isomeric Hydrocarbons of the Ethylene Series, J. Amer. Chem. Soc., 55 (2) (1933), 680-686.
F. Hermann, Ueber das Problem, die Anzahl der isomeren Paraffine der Formel C_n H_{2n+2} zu bestimmung, Chem. Ber., 13 (1880), 792. [I (1880), Annotated scanned copy]
F. Hermann, Ueber das Problem, die Anzahl der isomeren Paraffine der Formel C_n H_{2n+2} zu bestimmung, Chem. Ber., 30 (1897), 2423-2426. [II (1897), Annotated scanned copy]
F. Hermann, Entgegnung, Chem. Ber., 31 (1898), 91. [III (1898), Annotated scanned copy]
Lorentz Jäntschi and Lavinia-Lorena Pruteanu, Geometry of C_32 cyclic polyyne and some of its clusters, Carpathian J. Math. (2025) Vol. 41, No. 2, 371-391. See p. 371.
Michael A. Kappler, GENSMI: Exhaustive Enumeration of Simple Graphs. Daylight CIS, Inc., EuroMUG '04;4-Nov 05 2004.
E. V. Konstantinova and M. V. Vidyuk, Discriminating tests of information and topological indices; animals and trees; J. Chem. Inf. Comput. Sci., 43 (2003), 1860-1871.
J. Lederberg, Letter to N. J. A. Sloane, Aug 13 1971
J. Lederberg, The topology of molecules, in The Mathematical Sciences. Cambridge, Mass.: MIT Press, 1969, pages 37-51. [Annotated scanned copy]
J. Lederberg, Dendral-64, I, Report to NASA, Dec 1964 [Annotated scanned copy]
J. Lederberg, Dendral-64, II, Report to NASA, Dec 1965 [Annotated scanned copy]
J. Lederberg, Dendral-64, III, Report to NASA, Dec 1968 [Annotated scanned copy]
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1 (1992), pp. 53-80.
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
S. M. Losanitsch, Bemerkungen zu der Hermasnnschen Mitteillung: Die Anzahl..., Chem. Ber., 30 (1897), 3059-3060 [Annotated scanned copy]
A. Masoumi, M. Antoniazzi, and M. Soutchanski, Modeling organic chemistry and planning organic synthesis, GCAI 2015. Global Conference on Artificial Intelligence (2015), pp. 176-195.
W. R. Mueller et al., Molecular topological index, J. Chem. Inf. Comput. Sci., 30 (1990),160-163.
R. Otter, The number of trees, Ann. of Math. (2) 49 (1948), 583-599 discusses asymptotics.
D. Perry, The number of structural isomers of certain homologs of methane and methanol, J. Amer. Chem. Soc. 54 (1932), 2918-2920. [Gives a(60) correctly] (Annotated scanned copy)
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443. (Annotated scanned copy)
E. M. Rains and N. J. A. Sloane, On Cayley's Enumeration of Alkanes (or 4-Valent Trees), J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press, 1976. [Annotated scanned copy] See p. 28.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361.
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and achiral alkanes and monosubstituted alkanes, Tetrahedron 32 (3) (1976), 355-361. (Annotated scanned copy)
Hugo Schiff, Zur Statistik chemischer Verbindungen, Berichte der Deutschen Chemischen Gesellschaft, Vol. 8, pp. 1542-1547, 1875. [Annotated scanned copy]
Nicholas I. Shepherd-Barron, Asymptotic period relations for Jacobian elliptic surfaces, arXiv:1904.13344 [math.AG], 2019.
N. J. A. Sloane, Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Stephan Wagner, On the average Wiener index of degree-restricted trees
Sylvia Wenmackers, Huiswerk (met bijna twintig jaar vertraging) (in Dutch), 2015.
Index entries for sequences related to trees
Index entries for "core" sequences

Column k=4 of A144528.
A000602 = A000022 + A000200 for n>0.
Cf. A000598, A000625, A000628, A067608, A067609, A067610.

A000602 := proc(n)
    if n=0 then
        1
    else
        A000022(n)+A000200(n);
    end if;
end proc:

n = 40; (* algorithm from Rains and Sloane *)
S3[f_,h_,x_] := f[h,x]^3/6 + f[h,x] f[h,x^2]/2 + f[h,x^3]/3;
S4[f_,h_,x_] := f[h,x]^4/24 + f[h,x]^2 f[h,x^2]/4 + f[h,x] f[h,x^3]/3 + f[h,x^2]^2/8 + f[h,x^4]/4;
T[-1,z_] := 1;  T[h_,z_] := T[h,z] = Table[z^k, {k,0,n}].Take[CoefficientList[z^(n+1) + 1 + S3[T,h-1,z]z, z], n+1];
Sum[Take[CoefficientList[z^(n+1) + S4[T,h-1,z]z - S4[T,h-2,z]z - (T[h-1,z] - T[h-2,z]) (T[h-1,z]-1),z], n+1], {h,1,n/2}] + PadRight[{1,1}, n+1] + Sum[Take[CoefficientList[z^(n+1) + (T[h,z] - T[h-1,z])^2/2 + (T[h,z^2] - T[h-1,z^2])/2, z],n+1], {h,0,n/2}] (* Robert A. Russell, Sep 15 2018 *)
b[n_, i_, t_, k_] := b[n,i,t,k] = If[i<1, 0, Sum[Binomial[b[i-1,i-1,
  k,k] + j-1, j]* b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]];
b[0, i_, t_, k_] = 1; m = 3; (* m = maximum children *) n = 40;
gf[x_] = 1 + Sum[b[j-1,j-1,m,m]x^j,{j,1,n}]; (* G.f. for A000598 *)
ci[x_] = SymmetricGroupIndex[m+1, x] /. x[i_] -> gf[x^i];
CoefficientList[Normal[Series[gf[x] - (gf[x]^2 - gf[x^2])/2 + x ci[x],
{x, 0, n}]],x] (* Robert A. Russell, Jan 19 2023 *)

a(n) = A010372(n) + A010373(n/2) for n even, a(n) = A010372(n) for n odd.
Also equals A000022 + A000200 (n>0), both of which have known generating functions. Also g.f. = A000678(x) - A000599(x) + A000598(x^2) = (x + x^2 + 2x^3 + ...) - (x^2 + x^3 + 3x^4 + ...) + (1 + x^2 + x^4 + ...) = 1 + x + x^2 + x^3 + 2x^4 + 3x^5 + ...
G.f.: B(x) - cycle_index(S2,-B(x)) + x * cycle_index(S4,B(x)) = B(x) - (B(x)^2 - B(x^2)) / 2 + x * (B(x)^4 + 6*B(x)^2*B(x^2) + 8*B(x)*B(x^3) + 3*B(x^2)^2 + 6*B(x^4)) / 24, where B(x) = 1 + x * cycle_index(S3,B(x)) = 1 + x * (B(x)^3 + 3*B(x)*B(x^2) + 2*B(x^3)) / 6 is the generating function for A000598. - Robert A. Russell, Jan 16 2023

Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20 2003

A002727 Number of 3 X n binary matrices up to row and column permutations.

Original entry on oeis.org

1, 4, 13, 36, 87, 190, 386, 734, 1324, 2284, 3790, 6080, 9473, 14378, 21323, 30974, 44159, 61898, 85440, 116286, 156240, 207446, 272432, 354162, 456097, 582238, 737205, 926298, 1155567, 1431892, 1763074, 2157904, 2626276, 3179278, 3829294, 4590118, 5477081

Offset: 0

N. J. A. Sloane

Also, number of unlabeled bipartite graphs with three left vertices and n right vertices. - Yavuz Oruc, Jan 22 2018

			G.f. = 1 + 4*x + 13*x^2 + 36*x^3 + 87*x^4 + 190*x^5 + 386*x^6 + 734*x^7 + ...

A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
A. Atmaca, and A. Yavuz Oruç, On the size of two families of unlabeled bipartite graphs, AKCE International Journal of Graphs and Combinatorics, 2017.
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
M. A. Harrison, On the number of classes of binary matrices, IEEE Transactions on Computers, C-22.12 (1973), 1048-1052. (Annotated scanned copy)
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Index entries for sequences related to binary matrices
Index entries for linear recurrences with constant coefficients, signature (4, -4, -2, 2, 4, 3, -12, 3, 4, 2, -2, -4, 4, -1).
Index entries for two-way infinite sequences

Cf. A000601, A002623, A006148, A006381.

A row of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

I:=[1,4,13,36,87,190,386,734,1324,2284,3790,6080,9473, 14378]; [n le 14 select I[n] else 4*Self(n-1)-4*Self(n-2)-2*Self(n-3)+2*Self(n-4)+4*Self(n-5)+3*Self(n-6)-12*Self(n-7)+ 3*Self(n-8)+4*Self(n-9)+2*Self(n-10)-2*Self(n-11)-4*Self(n-12)+4*Self(n-13)-Self(n-14): n in [1..50]]; // Vincenzo Librandi, Oct 13 2015

CoefficientList[Series[(x^6+x^4+2x^3+x^2+1)/((1-x)^4(1-x^2)^2(1-x^3)^2),{x,0,40}],x] (* or *) LinearRecurrence[{4,-4,-2,2,4,3,-12,3,4,2,-2,-4,4,-1},{1,4,13,36,87,190,386,734,1324,2284,3790,6080,9473,14378},41] (* Harvey P. Dale, Nov 10 2011 *)
Table[Which[
Mod[n, 3] == 0,
1/6 (1/27 (54 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 1,
1/6 (1/27 (50 + 45 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7]),
Mod[n, 3] == 2,
1/6 (1/27 (28 + 39 n + 12 n^2 + n^3) + 1/320 (4 + n) *(225 + 15 (-1)^n + 352 n + 172 n^2 + 32 n^3 + 2 n^4) + Binomial[7 + n, 7])
], {n, 0, 100}] (* Yavuz Oruc, Jan 22 2018 *)

{a(n) = (6*n^7 + 168*n^6 + 2121*n^5 + 15540*n^4 + 70084*n^3 + 190512*n^2 + n*[284544, 281709, 277824, 281709, 284544, 274989][n%6+1]) \ 181440 + 1}; /* Michael Somos, Aug 22 2016 */

x='x+O('x^99); Vec((1+x^2+2*x^3+x^4+x^6)/((1-x)^2*((1-x)*(1-x^2)*(1-x^3))^2)) \\ Altug Alkan, Mar 03 2018

Vec(G(3, x) + O(x^40)) \\ G defined in A028657. - Andrew Howroyd, Feb 28 2023

G.f.: (x^6+x^4+2*x^3+x^2+1)/((1-x)^4*(1-x^2)^2*(1-x^3)^2). - Vladeta Jovovic, Feb 04 2000.
a(0)=1, a(1)=4, a(2)=13, a(3)=36, a(4)=87, a(5)=190, a(6)=386, a(7)=734, a(8)=1324, a(9)=2284, a(10)=3790, a(11)=6080, a(12)=9473, a(13)=14378. For n>13, a(n)=4*a(n-1)-4*a(n-2)-2*a(n-3)+2*a(n-4)+4*a(n-5)+3*a(n-6)- 12*a(n-7)+ 3*a(n-8)+4*a(n-9)+2*a(n-10)-2*a(n-11)-4*a(n-12)+4*a(n-13)-a(n-14). - Harvey P. Dale, Nov 10 2011
a(n) = -a(-8 - n) for all n in Z. - Michael Somos, Aug 22 2016
From Yavuz Oruc, Jan 22 2018: (Start)
If n == 0 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 54))/54).
If n == 1 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 45n + 50))/54).
If n == 2 (mod 3) then a(n)=(1/6)*(binomial(n+7,7) + (3(n+4)(2n^4 + 32n^3 + 172n^2 + 352n + 15(-1)^n + 225))/960 + (2(n^3 + 12n^2 + 39n + 28))/54). (End)

More terms from Vladeta Jovovic, Feb 04 2000

Definition corrected by Max Alekseyev, Feb 05 2010

A002624 Expansion of (1-x)^(-3) * (1-x^2)^(-2).

Original entry on oeis.org

1, 3, 8, 16, 30, 50, 80, 120, 175, 245, 336, 448, 588, 756, 960, 1200, 1485, 1815, 2200, 2640, 3146, 3718, 4368, 5096, 5915, 6825, 7840, 8960, 10200, 11560, 13056, 14688, 16473, 18411, 20520, 22800, 25270, 27930, 30800, 33880, 37191, 40733, 44528

Offset: 0

N. J. A. Sloane

Given an irregular triangular matrix M with the triangular numbers in every column shifted down twice for columns >0, A002624 = M * [1, 2, 3, ...]. Example: row 4 of triangle M = (15, 6, 1), then (15, 6, 1) dot (1, 2, 3) = a(4) = 30 = (15 + 12 + 3). - Gary W. Adamson, Mar 02 2010
The Kn21, Kn22, Kn23, Fi2 and Ze2 triangle sums of A139600 are related to the sequence given above, e.g., Ze2(n) = a(n-1) - a(n-2) - a(n-3) + 4*a(n-4), with a(n) = 0 for n <= -1. For the definitions of these triangle sums see A180662. - Johannes W. Meijer, Apr 29 2011
8*a(n) + 16*a(n+1) + 16*a(n+2) is the number of ways to place 3 queens on an (n+6) X (n+6) chessboard so that they diagonally attack each other exactly twice. Also true for the nonexistent terms for n=-1, n=-2 and n=-3 assuming that they are zeros. In graph-theory representation they thus form the corresponding open walk (Eulerian trail) with V={1,2,3} vertices and length of 2. - Antal Pinter, Dec 31 2015
a(n) is the number of partitions of n into parts with three kinds of 1 and two kinds of 2. - Joerg Arndt, Jan 18 2016

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 20.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 204
Antal Pinter, Numerical solution of the k=3 Queens problem, 2011, Q(n) at p.8.
Antal Pinter, Software utility for enumerating positions of non-attacking and attacking chess pieces , Backtrack_V7Pro
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
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A047659, A139600, A180662.

[( (n+1)^4 +10*(n+1)^3 +32*(n+1)^2 +32*(n+1) +(6*(n+1) +15)*((n+1) mod 2) )/96 : n in [0..50]]; // Vincenzo Librandi, Oct 08 2011

A002624:=-1/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation

f[n_] := Block[{m = n - 1}, (m^4 + 10m^3 + 32m^2 + 32m + (6m + 15)Mod[m, 2])/96]; Table[ f[n], {n, 2, 45}]
(* Or *) CoefficientList[ Series[1/((1 - x)^3 (1 - x^2)^2), {x, 0, 44}], x] (* Robert G. Wilson v, Feb 26 2005 *)

Vec(1/(1-x)^3/(1-x^2)^2+O(x^99)) \\ Charles R Greathouse IV, Apr 19 2012

a(n)=(n^4 + 14*n^3 + 68*n^2 + 136*n - n%2*(6*n + 21))/96 + 1 \\ Charles R Greathouse IV, Feb 18 2016

a(n-1) = ( n^4 +10*n^3 +32*n^2 +32*n +(6*n +15)*(n mod 2) )/96.
From Antal Pinter, Oct 03 2014: (Start)
a(n) = C(n + 2, 2) + 2*C(n, 2) + 3*C(n - 2, 2) + 4*C(n - 4, 2) + ...
a(n) = Sum_{i = 1..z} i*C(n + 4 - 2i, 2)  where z = (2*n + 3 + (-1)^n)/4.
a(n) = (3*(2*n + 7)*(-1)^n + 2*n^4 + 28*n^3 + 136*n^2 + 266*n + 171)/192.
(End)
a(n) = A007009(n+1) - A001752(n-1) for n>0. - Antal Pinter, Dec 27 2015
a(n) = Sum_{j=0..n+1} A006918(j). - Richard Turk, Feb 18 2016

Formula and more terms from Frank Ellermann, Mar 14 2002

A057524 Number of 3 x n binary matrices without unit columns up to row and column permutations.

Original entry on oeis.org

1, 3, 7, 14, 25, 41, 64, 95, 136, 189, 256, 339, 441, 564, 711, 885, 1089, 1326, 1600, 1914, 2272, 2678, 3136, 3650, 4225, 4865, 5575, 6360, 7225, 8175, 9216, 10353, 11592, 12939, 14400, 15981, 17689, 19530, 21511, 23639, 25921, 28364, 30976

Offset: 0

Vladeta Jovovic, Sep 02 2000

Unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 3 points of that set uniquely (if offset is 3).

			There are 7 binary 3x2 matrices without unit columns up to row and column permutations:
[0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 0] [1 1] [1 1]
[0 0] [0 1] [1 1] [0 1] [1 1] [1 1] [1 1].

Author?, Table of n, a(n) for n = 0..1374
Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,0,1,2,-3,1).

Cf. A038846 for labeled case.

Cf. A000217, A002623, A002620.

CoefficientList[ Series[ 1/(1 - x^3)/(1 - x^2)/(1 - x)^3, {x, 0, 42}], x] (* Jean-François Alcover, Mar 26 2013 *)

(1/6)*(Z(S_n; 5, 5, ...)+3*Z(S_n; 3, 5, 3, 5, ...)+2*Z(S_n; 2, 2, 5, 2, 2, 5, ...)) where Z(S_n; x_1, x_2, x_3, ...) is cycle index of symmetric group S_n of degree n.
G.f.: 1/(1-x^3)/(1-x^2)/(1-x)^3.
Let P(i,k) be the number of integer partitions of n into k parts, then with k=3 we have a(n) = Sum_{m=1..n} Sum_{i=k..m} P(i,k). - Thomas Wieder, Feb 18 2007
a(n) = Sum_{m=0..n} (n-m+1)*floor(((m+3)^2+3)/12). [Renzo Benedetti, Sep 30 2009]
a(n) = floor( ((n+2)*(n+6)/12)^2 ) = round( ((n+2)*(n+6)/12)^2 ). [Renzo Benedetti, Jul 25 2012]
Partial sums of A000601. - R. J. Mathar, Jul 25 2012

More terms from James Sellers, Sep 07 2000

A062890 Number of quadrilaterals that can be formed with perimeter n. In other words, number of partitions of n into four parts such that the sum of any three is more than the fourth.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 8, 11, 12, 16, 18, 23, 24, 31, 33, 41, 43, 53, 55, 67, 69, 83, 86, 102, 104, 123, 126, 147, 150, 174, 177, 204, 207, 237, 241, 274, 277, 314, 318, 358, 362, 406, 410, 458, 462, 514, 519, 575, 579, 640, 645, 710

Offset: 0

Amarnath Murthy, Jun 29 2001

Partition sets of n into four parts (sides) such that the sum of any three is more than the fourth do not uniquely define a quadrilateral, even if it is further constrained to be cyclic. This is because the order of adjacent sides is important. E.g. the partition set [1,1,2,2] for a perimeter n=6 can be reordered to generate two non-congruent cyclic quadrilaterals, [1,2,1,2] and [1,1,2,2], where the first is a rectangle and the second a kite. - Frank M Jackson, Jun 29 2012

			a(7) = 2 as the two partitions are (1,2,2,2), (1,1,2,3) and in each sum of any three is more than the fourth.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
G. E. Andrews, P. Paule and A. Riese, MacMahon's partition analysis III. The Omega package, p. 19.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-gon partitions, Bull. Austral Math. Soc., 64 (2001), 321-329.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,1,-1,0,0,-1,1,-1,1,1,-1).

Number of k-gons that can be formed with perimeter n: A005044 (k=3), this sequence (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

CoefficientList[Series[x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)), {x, 0, 60}], x] (* Frank M Jackson, Jun 09 2017 *)

G.f.: x^4*(1+x+x^5)/((1-x^2)*(1-x^3)*(1-x^4)*(1-x^6)).
a(2*n+6) = A026810(2*n+6) - A000601(n), a(2*n+7) = A026810(2*n+7) - A000601(n) for n >= 0. - Seiichi Manyama, Jun 08 2017
From Wesley Ivan Hurt, Jan 01 2021: (Start)
a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-8) + a(n-9) - a(n-10) + a(n-11) + a(n-12) - a(n-13).
a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} sign(floor((i+j+k)/(n-i-j-k+1))). (End)

More terms from Vladeta Jovovic and Dean Hickerson, Jul 01 2001

A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94, 121, 155, 194, 241, 295, 359, 431, 515, 609, 717, 837, 973, 1123, 1292, 1477, 1683, 1908, 2157, 2427, 2724, 3045, 3396, 3774, 4185, 4626, 5104, 5615, 6166, 6754, 7386, 8058, 8778, 9542, 10358, 11222, 12142, 13114

Offset: 0

N. J. A. Sloane

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).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.
E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 199
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
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1).

Partial sums of A001400. Column 4 of A092905.

A002621:=-1/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**5; # Simon Plouffe in his 1992 dissertation
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r+2), right=Set(U, card=1)}, unlabeled]: subs(r=2, stack): seq(count(subs(r=2, ZL), size=m), m=4..51) ; # Zerinvary Lajos, Feb 07 2008
A057077 := proc(n) (-1)^floor(n/2) ; end proc:
A061347 := proc(n) op(1+(n mod 3),[1,1,-2]) ; end proc:
A002621 := proc(n) 83/288*n^2+55/64*n+2815/3456+11/288*n^3+1/576*n^4+11/128*(-1)^n+1/64*(-1)^n*n; %+ A057077(n)/16 +A061347(n)/27; end proc:
seq(A002621(n),n=0..10) ; # R. J. Mathar, Mar 15 2011

CoefficientList[Series[1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)),{x,0,60}],x] (* Stefan Steinerberger, Jun 10 2007 *)
LinearRecurrence[{2, 0, -1, 0, -2, 2, 0, 1, 0, -2, 1}, {1, 2, 4, 7, 12, 18, 27, 38, 53, 71, 94}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)

a(n)=(n+1)*(9*(-1)^n+n^3+21*n^2+145*n+350)\/576 \\ Charles R Greathouse IV, May 23 2013

a(n) = +2*a(n-1) - a(n-3) - 2*a(n-5) + 2*a(n-6) + a(n-8) - 2*a(n-10) + a(n-11).
a(n) = 83*n^2/288 +55*n/64 +2815/3456 +11*n^3/288 +n^4/576 +11*(-1)^n/128 +(-1)^n*n/64 + A057077(n)/16 +A061347(n)/27. - R. J. Mathar, Mar 15 2011
a(n)=floor((n+1)*(9*(-1)^n + n^3 + 21*n^2 + 145*n + 350)/576 + 1/2). - Tani Akinari, Nov 10 2012

Showing 1-10 of 26 results. Next

cp's OEIS Frontend

A002620 Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and w*x < 2*y*z.

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A001399 a(n) is the number of partitions of n into at most 3 parts; also partitions of n+3 in which the greatest part is 3; also number of unlabeled multigraphs with 3 nodes and n edges.

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A002623 Expansion of 1/((1-x)^4*(1+x)).

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A000602 Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.

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A211795 Number of (w,x,y,z) with all terms in {1,...,n} and wx < 2y*z.

A002621 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)*(1-x^4)).