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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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)

References

  • Sergei Abramovich, Combinatorics of the Triangle Inequality: From Straws to Experimental Mathematics for Teachers, Spreadsheets in Education (eJSiE), Vol. 9, Issue 1, Article 1, 2016. See Fig. 3.
  • G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27th Competition.
  • T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 73, problem 25.
  • Michael Doob, The Canadian Mathematical Olympiad -- L'Olympiade Mathématique du Canada 1969-1993, Canadian Mathematical Society -- Société Mathématique du Canada, Problème 9, 1970, pp 22-23, 1993.
  • H. J. Finck, On the chromatic numbers of a graph and its complement. Theory of Graphs (Proc. Colloq., Tihany, 1966) Academic Press, New York (1968), 99-113.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 99.
  • D. E. Knuth, The art of programming, Vol. 1, 3rd Edition, Addison-Wesley, 1997, Ex. 36 of section 1.2.4.
  • J. Nelder, Critical sets in Latin squares, CSIRO Division of Math. and Stats. Newsletter, Vol. 38 (1977), p. 4.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

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).

Programs

  • GAP
    # 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
  • Haskell
    a002620 = (`div` 4) . (^ 2) -- Reinhard Zumkeller, Feb 24 2012
  • Magma
    [ Floor(n/2)*Ceiling(n/2) : n in [0..40]];
  • Maple
    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
  • Mathematica
    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 *)
  • Maxima
    makelist(floor(n^2/4),n,0,50); /* Martin Ettl, Oct 17 2012 */
  • PARI
    a(n)=n^2\4
  • PARI
    (t(n)=n*(n+1)/2);for(i=1,50,print1(",",(-1)^i*sum(k=1,i,(-1)^k*t(k))))
  • PARI
    a(n)=n^2>>2 \\ Charles R Greathouse IV, Nov 11 2009
  • PARI
    x='x+O('x^100); concat([0, 0], Vec(x^2/((1-x)^2*(1-x^2)))) \\ Altug Alkan, Oct 15 2015
  • Python
    def A002620(n): return (n**2)>>2 # Chai Wah Wu, Jul 07 2022
  • Sage
    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

Formula

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

A004526 Nonnegative integers repeated, floor(n/2).

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of elements in the set {k: 1 <= 2k <= n}.
Dimension of the space of weight 2n+4 cusp forms for Gamma_0(2).
Dimension of the space of weight 1 modular forms for Gamma_1(n+1).
Number of ways 2^n is expressible as r^2 - s^2 with s > 0. Proof: (r+s) and (r-s) both should be powers of 2, even and distinct hence a(2k) = a(2k-1) = (k-1) etc. - Amarnath Murthy, Sep 20 2002
Lengths of sides of Ulam square spiral; i.e., lengths of runs of equal terms in A063826. - Donald S. McDonald, Jan 09 2003
Number of partitions of n into two parts. A008619 gives partitions of n into at most two parts, so A008619(n) = a(n) + 1 for all n >= 0. Partial sums are A002620 (Quarter-squares). - Rick L. Shepherd, Feb 27 2004
a(n+1) is the number of 1's in the binary expansion of the Jacobsthal number A001045(n). - Paul Barry, Jan 13 2005
Number of partitions of n+1 into two distinct (nonzero) parts. Example: a(8) = 4 because we have [8,1],[7,2],[6,3] and [5,4]. - Emeric Deutsch, Apr 14 2006
Complement of A000035, since A000035(n)+2*a(n) = n. Also equal to the partial sums of A000035. - Hieronymus Fischer, Jun 01 2007
Number of binary bracelets of n beads, two of them 0. For n >= 2, a(n-2) is the number of binary bracelets of n beads, two of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008
Let A be the Hessenberg n X n matrix defined by: A[1,j] = j mod 2, A[i,i]:=1, A[i,i-1] = -1, and A[i,j] = 0 otherwise. Then, for n >= 1, a(n+1) = (-1)^n det(A). - Milan Janjic, Jan 24 2010
From Clark Kimberling, Mar 10 2011: (Start)
Let RT abbreviate rank transform (A187224). Then
RT(this sequence) = A187484;
RT(this sequence without 1st term) = A026371;
RT(this sequence without 1st 2 terms) = A026367;
RT(this sequence without 1st 3 terms) = A026363. (End)
The diameter (longest path) of the n-cycle. - Cade Herron, Apr 14 2011
For n >= 3, a(n-1) is the number of two-color bracelets of n beads, three of them are black, having a diameter of symmetry. - Vladimir Shevelev, May 03 2011
Pelesko (2004) refers erroneously to this sequence instead of A008619. - M. F. Hasler, Jul 19 2012
Number of degree 2 irreducible characters of the dihedral group of order 2(n+1). - Eric M. Schmidt, Feb 12 2013
For n >= 3 the sequence a(n-1) is the number of non-congruent regions with infinite area in the exterior of a regular n-gon with all diagonals drawn. See A217748. - Martin Renner, Mar 23 2013
a(n) is the number of partitions of 2n into exactly 2 even parts. a(n+1) is the number of partitions of 2n into exactly 2 odd parts. This just rephrases the comment of E. Deutsch above. - Wesley Ivan Hurt, Jun 08 2013
Number of the distinct rectangles and square in a regular n-gon is a(n/2) for even n and n >= 4. For odd n, such number is zero, see illustration in link. - Kival Ngaokrajang, Jun 25 2013
x-coordinate from the image of the point (0,-1) after n reflections across the lines y = n and y = x respectively (alternating so that one reflection is applied on each step): (0,-1) -> (0,1) -> (1,0) -> (1,2) -> (2,1) -> (2,3) -> ... . - Wesley Ivan Hurt, Jul 12 2013
a(n) is the number of partitions of 2n into exactly two distinct odd parts. a(n-1) is the number of partitions of 2n into exactly two distinct even parts, n > 0. - Wesley Ivan Hurt, Jul 21 2013
a(n) is the number of permutations of length n avoiding 213, 231 and 312, or avoiding 213, 312 and 321 in the classical sense which are breadth-first search reading words of increasing unary-binary trees. For more details, see the entry for permutations avoiding 231 at A245898. - Manda Riehl, Aug 05 2014
Also a(n) is the number of different patterns of 2-color, 2-partition of n. - Ctibor O. Zizka, Nov 19 2014
Minimum in- and out-degree for a directed K_n (see link). - Jon Perry, Nov 22 2014
a(n) is also the independence number of the triangular graph T(n). - Luis Manuel Rivera Martínez, Mar 12 2015
For n >= 3, a(n+4) is the least positive integer m such that every m-element subset of {1,2,...,n} contains distinct i, j, k with i + j = k (equivalently, with i - j = k). - Rick L. Shepherd, Jan 24 2016
More generally, the ordinary generating function for the integers repeated k times is x^k/((1 - x)(1 - x^k)). - Ilya Gutkovskiy, Mar 21 2016
a(n) is the number of numbers of the form F(i)*F(j) between F(n+3) and F(n+4), where 2 < i < j and F = A000045 (Fibonacci numbers). - Clark Kimberling, May 02 2016
The arithmetic function v_2(n,2) as defined in A289187. - Robert Price, Aug 22 2017
a(n) is also the total domination number of the (n-3)-gear graph. - Eric W. Weisstein, Apr 07 2018
Consider the numbers 1, 2, ..., n; a(n) is the largest integer t such that these numbers can be arranged in a row so that all consecutive terms differ by at least t. Example: a(6) = a(7) = 3, because of respectively (4, 1, 5, 2, 6, 3) and (1, 5, 2, 6, 3, 7, 4) (see link BMO - Problem 2). - Bernard Schott, Mar 07 2020
a(n-1) is also the number of integer-sided triangles whose sides a < b < c are in arithmetic progression with a middle side b = n (see A307136). Example, for b = 4, there exists a(3) = 1 such triangle corresponding to Pythagorean triple (3, 4, 5). For the triples, miscellaneous properties and references, see A336750. - Bernard Schott, Oct 15 2020
For n >= 1, a(n-1) is the greatest remainder on division of n by any k in 1..n. - David James Sycamore, Sep 05 2021
Number of incongruent right triangles that can be formed from the vertices of a regular n-gon is given by a(n/2) for n even. For n odd such number is zero. For a regular n-gon, the number of incongruent triangles formed from its vertices is given by A069905(n). The number of incongruent acute triangles is given by A005044(n). The number of incongruent obtuse triangles is given by A008642(n-4) for n > 3 otherwise 0, with offset 0. - Frank M Jackson, Nov 26 2022
The inverse binomial transform is 0, 0, 1, -2, 4, -8, 16, -32, ... (see A122803). - R. J. Mathar, Feb 25 2023

Examples

			G.f. = x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ...

References

  • G. L. Alexanderson et al., The William Powell Putnam Mathematical Competition - Problems and Solutions: 1965-1984, M.A.A., 1985; see Problem A-1 of 27th Competition.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, P(n,2).
  • Graham, Knuth and Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1989, page 77 (partitions of n into at most 2 parts).

Links

Crossrefs

a(n+2) = A008619(n). See A008619 for more references.
A001477(n) = a(n+1)+a(n). A000035(n) = a(n+1)-A002456(n).
a(n) = A008284(n, 2), n >= 1.
Zero followed by the partial sums of A000035.
Column 2 of triangle A094953. Second row of A180969.
Cf. A002264, A002265, A002266, A010761, A010762, A110532, A110533.
Partial sums: A002620. Other related sequences: A010872, A010873, A010874.
Cf. similar sequences of the integers repeated k times: A001477 (k = 1), this sequence (k = 2), A002264 (k = 3), A002265 (k = 4), A002266 (k = 5), A152467 (k = 6), A132270 (k = 7), A132292 (k = 8), A059995 (k = 10).
Cf. A289187, A139756 (binomial transf).
Cf. A307136, A336750.

Programs

  • Haskell
    a004526 = (`div` 2)
a004526_list = concatMap (\x -> [x, x]) [0..]
-- Reinhard Zumkeller, Jul 27 2012
  • Magma
    [Floor(n/2): n in [0..100]]; // Vincenzo Librandi, Nov 19 2014
  • Maple
    A004526 := n->floor(n/2); seq(floor(i/2),i=0..50);
  • Mathematica
    Table[(2n - 1)/4 + (-1)^n/4, {n, 0, 70}] (* Stefan Steinerberger, Apr 02 2006 *)
f[n_] := If[OddQ[n], (n - 1)/2, n/2]; Array[f, 74, 0] (* Robert G. Wilson v, Apr 20 2012 *)
With[{c=Range[0,40]},Riffle[c,c]] (* Harvey P. Dale, Aug 26 2013 *)
CoefficientList[Series[x^2/(1 - x - x^2 + x^3), {x, 0, 75}], x] (* Robert G. Wilson v, Feb 05 2015 *)
LinearRecurrence[{1, 1, -1}, {0, 0, 1}, 75] (* Robert G. Wilson v, Feb 05 2015 *)
Floor[Range[0, 40]/2] (* Eric W. Weisstein, Apr 07 2018 *)
  • Maxima
    makelist(floor(n/2),n,0,50); /* Martin Ettl, Oct 17 2012 */
  • PARI
    a(n)=n\2 /* Jaume Oliver Lafont, Mar 25 2009 */
  • PARI
    x='x+O('x^100); concat([0, 0], Vec(x^2/((1+x)*(x-1)^2))) \\ Altug Alkan, Mar 21 2016
  • Python
    def a(n): return n//2
print([a(n) for n in range(74)]) # Michael S. Branicky, Apr 30 2022
  • Sage
    def a(n) : return( dimension_cusp_forms( Gamma0(2), 2*n+4) ); # Michael Somos, Jul 03 2014
  • Sage
    def a(n) : return( dimension_modular_forms( Gamma1(n+1), 1) ); # Michael Somos, Jul 03 2014

Formula

G.f.: x^2/((1+x)*(x-1)^2).
a(n) = floor(n/2).
a(n) = ceiling((n+1)/2). - Eric W. Weisstein, Jan 11 2024
a(n) = 1 + a(n-2).
a(n) = a(n-1) + a(n-2) - a(n-3).
a(2*n) = a(2*n+1) = n.
a(n+1) = n - a(n). - Henry Bottomley, Jul 25 2001
For n > 0, a(n) = Sum_{i=1..n} (1/2)/cos(Pi*(2*i-(1-(-1)^n)/2)/(2*n+1)). - Benoit Cloitre, Oct 11 2002
a(n) = (2*n-1)/4 + (-1)^n/4; a(n+1) = Sum_{k=0..n} k*(-1)^(n+k). - Paul Barry, May 20 2003
E.g.f.: ((2*x-1)*exp(x) + exp(-x))/4. - Paul Barry, Sep 03 2003
G.f.: (1/(1-x)) * Sum_{k >= 0} t^2/(1-t^4) where t = x^2^k. - Ralf Stephan, Feb 24 2004
a(n+1) = A000120(A001045(n)). - Paul Barry, Jan 13 2005
a(n) = (n-(1-(-1)^n)/2)/2 = (1/2)*(n-|sin(n*Pi/2)|). Likewise: a(n) = (n-A000035(n))/2. Also: a(n) = Sum_{k=0..n} A000035(k). - Hieronymus Fischer, Jun 01 2007
The expression floor((x^2-1)/(2*x)) (x >= 1) produces this sequence. - Mohammad K. Azarian, Nov 08 2007; corrected by M. F. Hasler, Nov 17 2008
a(n+1) = A002378(n) - A035608(n). - Reinhard Zumkeller, Jan 27 2010
a(n+1) = A002620(n+1) - A002620(n) = floor((n+1)/2)*ceiling((n+1)/2) - floor(n^2/4). - Jonathan Vos Post, May 20 2010
For n >= 2, a(n) = floor(log_2(2^a(n-1) + 2^a(n-2))). - Vladimir Shevelev, Jun 22 2010
a(n) = A180969(2,n). - Adriano Caroli, Nov 24 2010
A001057(n-1) = (-1)^n*a(n), n > 0. - M. F. Hasler, Jul 19 2012
a(n) = A008615(n) + A002264(n). - Reinhard Zumkeller, Apr 28 2014
Euler transform of length 2 sequence [1, 1]. - Michael Somos, Jul 03 2014

Extensions

Partially edited by Joerg Arndt, Mar 11 2010, and M. F. Hasler, Jul 19 2012

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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

References

  • R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963; Chapter III, Problem 33.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 110, D(n); page 263, #18, P_n^{3}.
  • J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 517.
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, (4.1.18).
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
  • R. Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
  • J. H. van Lint, Combinatorial Seminar Eindhoven, Lecture Notes Math., 382 (1974), see pp. 33-34.
  • G. Pólya and G. Szegő, Problems and Theorems in Analysis I (Springer 1924, reprinted 1972), Part One, Chap. 1, Sect. 1, Problem 25.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

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.

Programs

  • Haskell
    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
  • Magma
    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
  • Magma
    [#RestrictedPartitions(n,{1,2,3}): n in [0..62]]; // Marius A. Burtea, Jan 06 2019
  • Magma
    [Round((n+3)^2/12): n in [0..70]]; // Marius A. Burtea, Jan 06 2019
  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    {a(n) = round((n + 3)^2 / 12)}; /* Michael Somos, Sep 04 2006 */
  • Python
    [round((n+3)**2 / 12) for n in range(0,62)] # Ya-Ping Lu, Jan 24 2024

Formula

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

Extensions

Name edited by Gus Wiseman, Apr 15 2019

A069905 Number of partitions of n into 3 positive parts.

Original entry on oeis.org

0, 0, 0, 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

Views

Author

N. J. A. Sloane, May 04 2002

Keywords

Comments

Number of binary bracelets of n beads, 3 of them 0. For n >= 3, a(n-3) is the number of binary bracelets of n beads, 3 of them 0, with 00 prohibited. - Washington Bomfim, Aug 27 2008
Also number of partitions of n-3 into parts 1, 2, and 3. - Joerg Arndt, Sep 05 2013
Number of incongruent triangles with integer sides that have perimeter 2n-3 (see the Jordan et al. link). - Freddy Barrera, Aug 18 2018
Number of ordered triples (x,y,z) of nonnegative integers such that x+y+z=n and xDennis P. Walsh, Apr 19 2019
Number of incongruent triangles formed from any 3 vertices of a regular n-gon. - Frank M Jackson, Sep 11 2022
Also a(n-3) for n > 2, otherwise 0 is the number of incongruent scalene triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 27 2022

Examples

			G.f. = x^3 + x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 5*x^8 + 7*x^9 + 8*x^10 + 10*x^11 + ...

References

  • Ross Honsberger, Mathematical Gems III, Math. Assoc. Amer., 1985, p. 39.
  • Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.4, p. 410.
  • Donald E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Links

Crossrefs

Another version of A001399, which is the main entry for this sequence.
Cf. A005044, A008284, A008615, A026810 (4 positive parts).
Cf. A198442, A079978, A000212, A002620, A008133, A307018.

Programs

  • GAP
    List([0..70],n->NrPartitions(n,3)); # Muniru A Asiru, May 17 2018
  • Haskell
    a069905 n = a069905_list !! n
a069905_list = scanl (+) 0 a008615_list
-- Reinhard Zumkeller, Apr 28 2014
  • Magma
    [(n^2+6) div 12: n in [0..70]]; // Vincenzo Librandi, Oct 14 2015
  • Maple
    A069905 := n->round(n^2/12): seq(A069905(n), n=0..70);
  • Mathematica
    a[ n_]:= Round[ n^2 / 12] (* Michael Somos, Sep 04 2013 *)
CoefficientList[Series[x^3/((1-x)(1-x^2)(1-x^3)), {x, 0, 70}], x] (* Vincenzo Librandi, Oct 14 2015 *)
Drop[LinearRecurrence[{1,1,0,-1,-1,1}, Append[Table[0,{5}],1],70],2] (* Robert A. Russell, May 17 2018 *)
  • PARI
    a(n) = floor((n^2+6)/12);  \\ Washington Bomfim, Jul 03 2012
  • PARI
    my(x='x+O('x^70)); concat([0, 0, 0], Vec(x^3/((1-x)*(1-x^2)*(1-x^3)))) \\ Altug Alkan, Oct 14 2015
  • SageMath
    [round(n^2/12) for n in range(70)] # G. C. Greubel, Apr 03 2019

Formula

G.f.: x^3/((1-x)*(1-x^2)*(1-x^3)) = x^3/((1-x)^3*(1+x+x^2)*(1+x)).
a(n) = round(n^2/12).
a(n) = floor((n^2+6)/12). - Washington Bomfim, Jul 03 2012
a(-n) = a(n). - Michael Somos, Sep 04 2013
a(n) = a(n-1) + A008615(n-1) for n > 0. - Reinhard Zumkeller, Apr 28 2014
Let n = 6k + m. Then a(n) = n^2/12 + a(m) - m^2/12. Also, a(n) = 3*k^2 + m*k + a(m). Example: a(35) = a(6*5 + 5) = 35^2/12 + a(5) - 5^2/12 = 102 = 3*5^2 + 5*5 + a(5). - Gregory L. Simay, Oct 13 2015
a(n) = a(n-1) +a(n-2) -a(n-4) -a(n-5) +a(n-6), n>5. - Wesley Ivan Hurt, Oct 16 2015
a(n) = A008284(n,3). - Robert A. Russell, May 13 2018
a(n) = A005044(2*n) = A005044(2*n - 3). - Freddy Barrera, Aug 18 2018
a(n) = floor((n^2+k)/12) for all integers k such that 3 <= k <= 7. - Giacomo Guglieri, Apr 03 2019
From Wesley Ivan Hurt, Apr 19 2019: (Start)
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} 1.
a(n) = Sum_{i=1..floor(n/3)} floor((n-i)/2) - i + 1. (End)
Sum_{n>=3} 1/a(n) = 15/4 + Pi^2/18 - Pi/(2*sqrt(3)) + tanh(Pi/(2*sqrt(3))) * Pi/sqrt(3). - Amiram Eldar, Sep 27 2022
E.g.f.: (8*exp(-x/2)*cos(sqrt(3)*x/2) + (3*x^2 + 3*x - 8)*cosh(x) + (3*x^2 + 3*x + 1)*sinh(x))/36. - Stefano Spezia, Apr 05 2023
From Ridouane Oudra, Dec 12 2024: (Start)
a(n) = (n^2 + 2*gcd(n,3) - 3*gcd(n,2))/12.
a(n) = (A198442(n) + A079978(n))/3.
a(n) = A000212(n) - A002620(n).
a(n) = A008133(n+1) - A307018(n+1). (End)
a(n) = (A309511(n) + A309513(n))/3. - Ray Chandler, Mar 13 2025

A008615 a(n) = floor(n/2) - floor(n/3).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 7, 6, 7, 7, 7, 7, 8, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 10, 9, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 12, 11, 12, 12, 12, 12, 13, 12, 13, 13, 13, 13, 14, 13, 14, 14, 14, 14, 15, 14
Offset: 0

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

If the two leading 0's are dropped, this becomes the essentially identical sequence A103221, with g.f. 1/((1-x^2)*(1-x^3)), which arises in many contexts. For example, 1/((1-x^4)*(1-x^6)) is the Poincaré series [or Poincare series] for modular forms of weight w for the full modular group. As generators one may take the Eisenstein series E_4 (A004009) and E_6 (A013973).
Dimension of the space of weight 2n+8 cusp forms for Gamma_0( 1 ).
Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 5 ).
a(n) is the number of ways n can be written as the sum of a positive even number and a nonnegative multiple of 3 and so the number of ways (n-2) can be written as the sum of a nonnegative even number and a nonnegative multiple of 3 and also the number of ways (n+3) can be written as the sum of a positive even number and a positive multiple of 3.
It appears that this is also the number of partitions of 2n+6 that are 4-term arithmetic progressions. - John W. Layman, May 01 2009 [verified by Wesley Ivan Hurt, Jan 17 2021]
a(n) is the number of (n+3)-digit fixed points under the base-3 Kaprekar map A164993 (see A164997 for the list of fixed points). - Joseph Myers, Sep 04 2009
Starting from n=10 also the number of balls in new consecutive hexagonal edges, if an (infinite) chain of balls is winded spirally around the first ball at the center, such that each six steps make an entire winding. - K. G. Stier, Dec 21 2012
In any three consecutive terms at least two of them are equal to each other. - Michael Somos, Mar 01 2014
Number of partitions of (n-2) into parts 2 and 3. - David Neil McGrath, Sep 05 2014
a(n), n >= 0, is also the dimension of S_{2*(n+4)}, the complex vector space of modular cusp forms of weight 2*(n+4) and level 1 (full modular group). The dimension of S_0, S_2, S_4 and S_6 is 0. See, e.g., Ash and Gross, p. 178. Table 13.1. - Wolfdieter Lang, Sep 16 2016
From Wolfdieter Lang, May 08 2017: (Start)
a(n-2) = floor((n-2)/2) - floor((n-2)/3) = floor(n/2) - floor((n+1)/3) is for n >=0 the number of integers k in the interval (n+1)/3 < k <= floor(n/2). This problem appears in the computation of the number of zeros of Chebyshev S(n, x) polynomials (coefficients in A049310) in the open interval (-1, +1). See a comment there. This computation was motivated by a conjecture given in A008611 by Michel Lagneau, Mar 31 2017.
a(n) is also the number of integers k in the closed interval (n+1)/3 <= k <= floor(n/2), which is floor(n/2) - (ceiling((n+1)/3) - 1) for n >= 0 (proof trivial for n+1 == 0 (mod 3) and otherwise). From the preceding statement this a(n) is also a(n-2) + [n == 2 (mod 3)] for n >= 0 (with [statement] = 1 if the statement is true and zero otherwise). This proves the recurrence given by Michael Somos in the formula section. (End)
Assuming the Collatz conjecture to be true, for n > 1, a(n+7) is the row length of the n-th row of A340985. That is, the number of weakly connected components of the Collatz digraph of order n. - Sebastian Karlsson, Feb 23 2021

Examples

			G.f. = x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + 2*x^12 + ...

References

  • Avner Ash and Robert Gross, Summing it up, Princeton University Press, 2016, p. 178.
  • D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
  • E. Freitag, Siegelsche Modulfunktionen, Springer-Verlag, Berlin, 1983; p. 141, Th. 1.1.
  • R. C. Gunning, Lectures on Modular Forms. Princeton Univ. Press, Princeton, NJ, 1962.
  • J.-M. Kantor, Où en sont les mathématiques, La formule de Molien-Weyl, SMF, Vuibert, p. 79

Links

Crossrefs

Essentially the same as A103221.
First differences of A069905 (and A001399).
Cf. A002264, A004009, A004526, A005044, A008588, A008611, A013973, A016921, A016933, A016957, A049310, A057078, A164993, A164997, A340985.

Programs

  • Haskell
    a008615 n = n `div` 2 - n `div` 3  -- Reinhard Zumkeller, Apr 28 2014
  • Magma
    [Floor(n/2)-Floor(n/3): n in [0..10]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006
  • Magma
    a := func< n | n lt 2 select 0 else n eq 2 select 1 else Dimension( ModularForms( PSL2( Integers()), 2*n-4))>; /* Michael Somos, Dec 11 2018 */
  • Maple
    a := n-> floor(n/2) - floor(n/3): seq(a(n), n = 0 .. 87);
  • Mathematica
    a[n_]:=Floor[n/2]-Floor[n/3]; Array[a,90,0] (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008; corrected by Harvey P. Dale, Nov 30 2011 *)
LinearRecurrence[{0, 1, 1, 0, -1}, {0, 0, 1, 0, 1}, 100]; (* Vincenzo Librandi, Sep 09 2015 *)
  • PARI
    {a(n) = (n\2) - (n\3)}; /* Michael Somos, Feb 06 2003 */
  • Python
    def A008615(n): return n//2 - n//3 # Chai Wah Wu, Jun 07 2022

Formula

a(n) = a(n-6) + 1 = a(n-2) + a(n-3) - a(n-5). - Henry Bottomley, Sep 02 2000
G.f.: x^2 / ((1-x^2) * (1-x^3)).
From Reinhard Zumkeller, Feb 27 2008: (Start)
a(A016933(n)) = a(A016957(n)) = a(A016969(n)) = n+1.
a(A008588(n)) = a(A016921(n)) = a(A016945(n)) = n. (End)
a(6*k) = k, k >= 0. - Zak Seidov, Sep 09 2012
a(n) = A005044(n+1) - A005044(n-3). - Johannes W. Meijer, Oct 18 2013
a(n) = floor((n+4)/6) - floor((n+3)/6) + floor((n+2)/6). - Mircea Merca, Nov 27 2013
Euler transform of length 3 sequence [0, 1, 1]. - Michael Somos, Mar 01 2014
a(n+2) = a(n) + 1 if n == 0 (mod 3), a(n+2) = a(n) otherwise. - Michael Somos, Mar 01 2014. See the May 08 2017 comment above. - Wolfdieter Lang, May 08 2017
a(n) = -a(-1 - n) for all n in Z. - Michael Somos, Mar 01 2014.
a(n) = A004526(n) - A002264(n). - Reinhard Zumkeller, Apr 28 2014
a(n) = Sum_{i=0..n-2} (floor(i/6)-floor((i-3)/6))*(-1)^i. - Wesley Ivan Hurt, Sep 08 2015
a(n) = a(n+6) - 1 = A103221(n+4) - 1, n >= 0. - Wolfdieter Lang, Sep 16 2016
12*a(n) = 2*n +1 +3*(-1)^n -4*A057078(n). - R. J. Mathar, Jun 19 2019
a(n) = Sum_{k=1..floor((n+3)/2)} Sum_{j=k..floor((2*n+6-k)/3)} Sum_{i=j..floor((2*n+6-j-k)/2)} ([j-k = i-j = 2*n+6-2*i-j-k] - [k = j = i = 2*n+6-i-j-k]), where [ ] is the (generalized) Iverson bracket. - Wesley Ivan Hurt, Jan 17 2021
E.g.f.: (3*(2 + x)*cosh(x) - 2*exp(-x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 3*(x-1)*sinh(x))/18. - Stefano Spezia, Oct 17 2022

A001400 Number of partitions of n into at most 4 parts.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350, 1425, 1495
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].
Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes. - Vladeta Jovovic, Dec 27 1999
Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002
a(n) is the coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002
Also number of partitions of n into parts <= 4. a(n) = A026820(n,4), for n > 3. - Reinhard Zumkeller, Jan 21 2010
Number of different distributions of n+10 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - Ece Uslu and Esin Becenen, Jan 11 2016
Number of partitions of 5n+8 or 5n+12 into 4 parts (+-) 3 mod 5. a(4) = 5 partitions of 28: [7,7,7,7], [12,7,7,2], [12,12,2,2], [17,7,2,2], [22,2,2,2]. a(3) = 3 partitions of 27: [8,8,8,3], [13,8,3,3], [18,3,3,3]. - Richard Turk, Feb 24 2016
a(n) is the total number of non-isomorphic geodetic graphs of diameter n homeomorphic to a complete graph K4. - Carlos Enrique Frasser, May 24 2018

Examples

			(4 choose 4)_q = 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 11*x^7 + ...
a(4) = 5, i.e., {1,2,3,8}, {1,2,4,7}, {1,2,5,6}, {2,3,4,5}, {1,3,4,6}. Number of different distributions of 14 identical balls in 4 boxes as x,y,z,p where 0 < x < y < z < p. - _Ece Uslu_, Esin Becenen, Jan 11 2016

References

  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).
  • H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.
  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
  • D. E. Knuth, The Art of Computer Programming, vol. 4, Fascicle 3, Generating All Combinations and Partitions, Addison-Wesley, 2005, Section 7.2.1.4., p. 56, exercise 31.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Essentially same as A026810. Partial sums of A005044.
a(n) = A008284(n+4, 4), n >= 0.
Cf. A008619, A001399, A001401, A026820, A070083, A117486.
First differences of A002621.

Programs

  • Haskell
    a001400 n = a001400_list !! n
a001400_list = scanl1 (+) a005044_list -- Reinhard Zumkeller, Feb 28 2013
  • Magma
    K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; G:=MatrixGroup<4,K|q1,q2,h>; MolienSeries(G);
  • Maple
    A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;
with(combstruct):ZL5:=[S,{S=Set(Cycle(Z,card<5))}, unlabeled]:seq(count(ZL5,size=n),n=0..55); # Zerinvary Lajos, Sep 24 2007
A001400:=-(-z**8+z**9+2*z**4-z**7-1-z)/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**4; # [conjectured by Simon Plouffe in his 1992 dissertation; gives sequence except for an initial 1]
B:=[S,{S = Set(Sequence(Z,1 <= card),card <=4)},unlabelled]: seq(combstruct[count](B, size=n), n=0..55); # Zerinvary Lajos, Mar 21 2009
  • Mathematica
    CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 3, 5, 6, 9, 11, 15, 18}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)
a[n_] := Sum[Floor[(n - j - 3*k + 2)/2], {j, 0, Floor[n/4]}, {k, j, Floor[(n - j)/3]}]; Table[a[n], {n, 0, 55}] (* L. Edson Jeffery, Jul 31 2014 *)
a[ n_] := With[{m = n + 5}, Round[ (2 m^3 - 3 m (5 + 3 (-1)^m)) / 288]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] Length[ IntegerPartitions[ m, 4]]]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := With[{m = Abs[n + 5] - 5}, Sign[n + 5] SeriesCoefficient[ 1 / ((1 - x) (1 - x^2) (1 - x^3) (1 - x^4)), {x, 0, m}]]; (* Michael Somos, Dec 29 2014 *)
Table[Length@IntegerPartitions[n, 4], {n, 0, 55}] (* Robert Price, Aug 18 2020 *)
  • PARI
    a(n) = round(((n+4)^3 + 3*(n+4)^2 -9*(n+4)*((n+4)% 2))/144) \\ Washington Bomfim, Jul 03 2012
  • PARI
    {a(n) = n+=5; round( (2*n^3 - 3*n*(5 + 3*(-1)^n)) / 288)}; \\ Michael Somos, Dec 29 2014
  • PARI
    a(n) = #partitions(n,,4); \\ Ruud H.G. van Tol, Jun 02 2024

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).
a(n) = 1 + (a(n-2) + a(n-3) + a(n-4)) - (a(n-5) + a(n-6) + a(n-7)) + a(n-9). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000
P(n, 4) = (1/288)*(2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n) - 32*pcr{1, -1, 0}(3, n) - 36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).
Let c(n) = Sum_{i=0..floor(n/3)} (1 + ceiling((n-3*i-1)/2)), then a(n) = Sum_{i=0..floor(n/4)} (1 + ceiling((n-4*i-1)/2) + c(n-4*i-3)). - Jon Perry, Jun 27 2003
Euler transform of finite sequence [1, 1, 1, 1].
(n choose 4)_q = (q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1)).
a(n) = round(((n+4)^3 + 3*(n+4)^2 - 9*(n+4)*((n+4) mod 2))/144). - Washington Bomfim, Jul 03 2012
a(n) = a(n-1) + a(n-2) - 2*a(n-5) + a(n-8) + a(n-9) - a(n-10). - David Neil McGrath, Sep 12 2014
a(n) = -a(-10-n) for all n in Z. - Michael Somos, Dec 29 2014
a(n) - a(n+1) - a(n+3) + a(n+4) = 0 if n is odd, else floor(n/4) + 2 for all n in Z. - Michael Somos, Dec 29 2014
a(n) = n^3/144 + n^2/24 - 7*n/144 + 1 + floor(n/4)/4 + floor(n/3)/3 + (n+5)*floor(n/2)/8 + floor((n+1)/4)/4. - Vaclav Kotesovec, Aug 18 2015
a(n) = a(n-4) + A001399(n). - Ece Uslu, Esin Becenen, Jan 11 2016, corrected Sep 25 2020
a(6*n) - a(6*n+1) - a(6*n+4) + a(6*n+5) = n+1. - Richard Turk, Apr 19 2016
a(n) = a(n-1) + A005044(n+3) for n>0, i.e., first differences is A005044. - Yuchun Ji, Oct 12 2020
From Vladimír Modrák and Zuzana Soltysova, Dec 09 2020: (Start)
a(n) = round((n + 3)^2/12) + Sum_{i=0..floor(n/4)} round((n - 4*i - 1)^2/12).
a(n) = floor(((n + 3)^2 + 4)/12) + Sum_{i=0..floor(n/4)} floor(((n - 4*i - 1)^2 + 4)/12). (End)
a(n) - a(n-3) = A008642(n). - R. J. Mathar, Jun 23 2021
a(n) - a(n-2) = A025767(n). - R. J. Mathar, Jun 23 2021
a(n) = round((2*n^3 + 30*n^2 + 135*n + 175)/288 + (-1)^n*(n+5)/32). - Dave Neary, Oct 28 2021
From Vladimír Modrák, Jul 13 2022: (Start)
a(n) = Sum_{j=0..floor(n/4)} Sum_{i=0..floor(n/3)} ceiling((max(0,n + 1 - 3*i - 4*j))/2).
a(n) = Sum_{i=0..floor(n/4)} floor(((n + 3 - 4*i)^2 + 4)/12). (End)
a(n) = floor(((n+4)^2*(n+7) - 9*(n+4)*(n mod 2) + 32)/144). - Vladimír Modrák, Mar 23 2025

A026810 Number of partitions of n in which the greatest part is 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Also number of partitions of n into exactly 4 parts.
Also the number of weighted cubic graphs on 4 nodes (=the tetrahedron) with weight n. - R. J. Mathar, Nov 03 2018
From Gus Wiseman, Jun 27 2021: (Start)
Also the number of strict integer partitions of 2n with alternating sum 4, or (by conjugation) partitions of 2n covering an initial interval of positive integers with exactly 4 odd parts. The strict partitions with alternating sum 4 are:
(4) (5,1) (6,2) (7,3) (8,4) (9,5) (10,6)
(5,2,1) (5,3,2) (5,4,3) (6,5,3) (7,6,3)
(6,3,1) (6,4,2) (7,5,2) (8,6,2)
(7,4,1) (8,5,1) (9,6,1)
(6,3,2,1) (6,4,3,1) (6,5,4,1)
(7,4,2,1) (7,4,3,2)
(7,5,3,1)
(8,5,2,1)
(6,4,3,2,1)
(End)

Examples

			From _Gus Wiseman_, Jun 27 2021: (Start)
The a(4) = 1 through a(10) = 9 partitions of length 4:
  (1111)  (2111)  (2211)  (2221)  (2222)  (3222)  (3322)
                  (3111)  (3211)  (3221)  (3321)  (3331)
                          (4111)  (3311)  (4221)  (4222)
                                  (4211)  (4311)  (4321)
                                  (5111)  (5211)  (4411)
                                          (6111)  (5221)
                                                  (5311)
                                                  (6211)
                                                  (7111)
(End)

References

  • G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.
  • D. E. Knuth, The Art of Computer Programming, vol. 4,fascicle 3, Generating All Combinations and Partitions, Section 7.2.1.4., p. 56, exercise 31.

Links

Crossrefs

Cf. A001400, A026811, A026812, A026813, A026814, A026815, A026816, A069905 (3 positive parts), A002621 (partial sums), A005044 (first differences).
A non-strict version is A000710 or A088218.
This is column k = 2 of A152146.
A reverse version is A343941.
Cf. A000041, A000070, A000097, A067659, A103919, A120452, A236559, A239830, A306145, A343942, A344616, A344649, A344651.

Programs

  • Magma
    [Round((n^3+3*n^2-9*n*(n mod 2))/144): n in [0..60]]; // Vincenzo Librandi, Oct 14 2015
  • Maple
    A049347 := proc(n)
        op(1+(n mod 3),[1,-1,0]) ;
end proc:
A056594 := proc(n)
        op(1+(n mod 4),[1,0,-1,0]) ;
end proc:
A026810 := proc(n)
        1/288*(n+1)*(2*n^2+4*n-13+9*(-1)^n) ;
        %-A049347(n)/9 ;
        %+A056594(n)/8 ;
end proc: # R. J. Mathar, Jul 03 2012
  • Mathematica
    Table[Count[IntegerPartitions[n], {4, _}], {n, 0, 60}]
LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {0, 0, 0, 0, 1, 1, 2, 3, 5, 6}, 60] (* Vincenzo Librandi, Oct 14 2015 *)
Table[Length[IntegerPartitions[n, {4}]], {n, 0, 60}] (* Eric Rowland, Mar 02 2017 *)
CoefficientList[Series[x^4/Product[1 - x^k, {k, 1, 4}], {x, 0, 60}], x] (* Robert A. Russell, May 13 2018 *)
  • PARI
    for(n=0, 60, print(n, " ", round((n^3 + 3*n^2 -9*n*(n % 2))/144))); \\ Washington Bomfim, Jul 03 2012
  • PARI
    x='x+O('x^60); concat([0, 0, 0, 0], Vec(x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)))) \\ Altug Alkan, Oct 14 2015
  • PARI
    vector(60, n, n--; (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 + real(I^n)/8 - ((n+2)%3-1)/9) \\ Altug Alkan, Oct 26 2015
  • PARI
    print1(0,", "); for(n=1,60,j=0;forpart(v=n,j++,,[4,4]); print1(j,", ")) \\ Hugo Pfoertner, Oct 01 2018

Formula

G.f.: x^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) = x^4/((1-x)^4*(1+x)^2*(1+x+x^2)*(1+x^2)).
a(n+4) = A001400(n). - Michael Somos, Apr 07 2012
a(n) = round( (n^3 + 3*n^2 -9*n*(n mod 2))/144 ). - Washington Bomfim, Jan 06 2021 and Jul 03 2012
a(n) = (n+1)*(2*n^2+4*n-13+9*(-1)^n)/288 -A049347(n)/9 +A056594(n)/8. - R. J. Mathar, Jul 03 2012
From Gregory L. Simay, Oct 13 2015: (Start)
a(n) = (n^3 + 3*n^2 - 9*n)/144 + a(m) - (m^3 + 3*m^2 - 9*m)/144 if n = 12k + m and m is odd. For example, a(23) = a(12*1 + 11) = (23^3 + 3*23^2 - 9*23)/144 + a(11) - (11^3 + 3*11^2 - 9*11)/144 = 94.
a(n) = (n^3 + 3*n^2)/144 + a(m) - (m^3 + 3*m^2)/144 if n = 12k + m and m is even. For example, a(22) = a(12*1 + 10) = (22^3 + 3*22^2)/144 + a(10) - (10^3 + 3*10^2)/144 = 84. (End)
a(n) = A008284(n,4). - Robert A. Russell, May 13 2018
From Gregory L. Simay, Jul 28 2019: (Start)
a(2n+1) = a(2n) + a(n+1) - a(n-3) and
a(2n) = a(2n-1) + a(n+2) - a(n-2). (End)

A266755 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^4)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 14, 19, 16, 21, 19, 24, 21, 27, 24, 30, 27, 33, 30, 37, 33, 40, 37, 44, 40, 48, 44, 52, 48, 56, 52, 61, 56, 65, 61, 70, 65, 75, 70, 80, 75, 85, 80, 91, 85, 96, 91, 102, 96, 108, 102, 114, 108, 120, 114, 127, 120, 133, 127, 140, 133, 147, 140, 154, 147, 161, 154, 169
Offset: 0

Views

Author

N. J. A. Sloane, Jan 10 2016

Keywords

Comments

This is the same as A005044 but without the three leading zeros. There are so many situations where one wants this sequence rather than A005044 that it seems appropriate for it to have its own entry.
But see A005044 (still the main entry) for numerous applications and references.
Also, Molien series for invariants of finite Coxeter group D_3.
The Molien series for the finite Coxeter group of type D_k (k >= 3) has g.f. = 1/Product_i (1-x^(1+m_i)) where the m_i are [1,3,5,...,2k-3,k-1]. If k is even only even powers of x appear, and we bisect the sequence.
Also, Molien series for invariants of finite Coxeter group A_3. The Molien series for the finite Coxeter group of type A_k (k >= 1) has g.f. = 1/Product_{i=2..k+1} (1-x^i). Note that this is the root system A_k not the alternating group Alt_k.
a(n) is the number of partitions of n into parts 2, 3, and 4. - Joerg Arndt, Apr 16 2017
From Gus Wiseman, May 23 2021: (Start)
Also the number of integer partitions of n into at most n/2 parts, none greater than 3. The case of any maximum is A110618. The case of any length is A001399. The Heinz numbers of these partitions are given by A344293.
For example, the a(2) = 1 through a(13) = 5 partitions are:
2 3 22 32 33 322 332 333 3322 3332 3333 33322
31 222 331 2222 3222 3331 32222 33222 33331
321 3221 3321 22222 33221 33321 322222
3311 32221 33311 222222 332221
33211 322221 333211
332211
333111
(End)

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 3*x^6 + 2*x^7 + 4*x^8 + ... - _Michael Somos_, Jan 29 2022

References

  • J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.1, page 59.

Links

Crossrefs

Molien series for finite Coxeter groups A_1 through A_12 are A059841, A103221, A266755, A008667, A037145, A001996, and A266776-A266781.
Molien series for finite Coxeter groups D_3 through D_12 are A266755, A266769, A266768, A003402, and A266770-A266775.
A variant of A005044.
Cf. A001400 (partial sums).
Cf. A308065.
Number of partitions of n whose Heinz number is in A344293.
A001399 counts partitions with all parts <= 3, ranked by A051037.
A025065 counts partitions of n with >= n/2 parts, ranked by A344296.
A035363 counts partitions of n with n/2 parts, ranked by A340387.
A110618 counts partitions of n into at most n/2 parts, ranked by A344291.
Cf. A000041, A008642, A279622, A325691, A344294, A344297.

Programs

  • Magma
    I:=[1,0,1,1,2,1,3,2,4]; [n le 9 select I[n] else Self(n-2)+ Self(n-3)+Self(n-4)-Self(n-5)-Self(n-6)-Self(n-7)+Self(n-9): n in [1..100]]; // Vincenzo Librandi, Jan 11 2016
  • Mathematica
    CoefficientList[Series[1/((1-x^2)(1-x^3)(1-x^4)), {x, 0, 100}], x] (* JungHwan Min, Jan 10 2016 *)
LinearRecurrence[{0,1,1,1,-1,-1,-1,0,1}, {1,0,1,1,2,1,3,2,4}, 100] (* Vincenzo Librandi, Jan 11 2016 *)
Table[Length[Select[IntegerPartitions[n],Length[#]<=n/2&&Max@@#<=3&]],{n,0,30}] (* Gus Wiseman, May 23 2021 *)
a[ n_] := Round[(n + 3*(2 - Mod[n,2]))^2/48]; (* Michael Somos, Jan 29 2022 *)
  • PARI
    Vec(1/((1-x^2)*(1-x^3)*(1-x^4)) + O(x^100)) \\ Michel Marcus, Jan 11 2016
  • PARI
    {a(n) = round((n + 3*(2-n%2))^2/48)}; /* Michael Somos, Jan 29 2022 */
  • Sage
    (1/((1-x^2)*(1-x^3)*(1-x^4))).series(x, 100).coefficients(x, sparse=False) # G. C. Greubel, Jun 13 2019

Formula

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>8. - Vincenzo Librandi, Jan 11 2016
a(n) = a(-9-n) for all n in Z. a(n) = a(n+3) for all n in 2Z. - Michael Somos, Jan 29 2022
E.g.f.: exp(-x)*(81 - 18*x + exp(2*x)*(107 + 60*x + 6*x^2) + 64*exp(x/2)*cos(sqrt(3)*x/2) + 36*exp(x)*(cos(x) - sin(x)))/288. - Stefano Spezia, Mar 05 2023
For n >= 3, if n is even, a(n) = a(n-3) + floor(n/4) + 1, otherwise a(n) = a(n-3). - Robert FERREOL, Feb 05 2024
a(n) = floor((n^2+9*n+(3*n+9)*(-1)^n+39)/48). - Hoang Xuan Thanh, Jun 03 2025

A059169 Number of partitions of n into 3 parts which form the sides of a nondegenerate isosceles triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 7, 8, 8, 9, 8, 9, 9, 10, 9, 10, 10, 11, 10, 11, 11, 12, 11, 12, 12, 13, 12, 13, 13, 14, 13, 14, 14, 15, 14, 15, 15, 16, 15, 16, 16, 17, 16, 17, 17, 18, 17, 18, 18, 19, 18, 19, 19, 20, 19, 20, 20
Offset: 1

Views

Author

Floor van Lamoen, Jan 13 2001

Keywords

Comments

Also number of 0's in n-th row of triangle in A071026. - Hans Havermann, May 26 2002
Exponent of 2 in factorization of A030436(n-1) and A026655(n-1). First differences of A001971. - Ralf Stephan, Mar 21 2004
Conjecture: this is 0 followed by A026922. - R. J. Mathar, Oct 05 2008 [See the g.f. given there by Michael Somos and the one given below for the proof. - Wolfdieter Lang, May 10 2017]
a(n+1) is for n >= 0 the number of integers k in the left-sided open interval ((n+1)/4, floor(n/2)]. This is needed for the number of zeros of Chebyshev S polynomials in the open interval (-sqrt(2), sqrt(2)) given in A285869. - Wolfdieter Lang, May 10 2017

Examples

			Consider the number 13. The following partitions give a nondegenerate triangle: 4 4 5; 3 5 5; 1 6 6; 2 5 6; 3 4 6. Since the first three partitions represent isosceles triangles, we have A059169(13) = 3.
G.f. = x^3 + x^5 + x^6 + 2*x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 2*x^12 + ...

Links

Crossrefs

Essentially the same as A008624.
Cf. A178804.

Programs

  • Haskell
    a059169 n = a059169_list !! (n-1)
a059169_list = map abs $ zipWith (-) (tail a178804_list) a178804_list
-- Reinhard Zumkeller, Nov 15 2014
  • Magma
    [Floor((n-1)/2) - Floor(n/4): n in [1..80]]; // G. C. Greubel, Mar 08 2018
  • Maple
    a[1] := 0: a[2] := 0: a[3] := 1: a[4] := 0: a[5] := 1: for n from 6 to 300 do a[n] := a[n-1] + a[n-4] - a[n-5]: end do: seq(a[n], n=1..82);
a := n -> A005044(n) - A005044(n-6): A005044 := n-> floor((1/48)*(n^2 + 3*n + 21 + (-1)^(n-1)*3*n)): seq(a(n), n = 1..82); # Johannes W. Meijer, Oct 10 2013
  • Mathematica
    CoefficientList[Series[x^2 (1 - x + x^2)/(1 - x - x^4 + x^5), {x, 0, 100}], x] (* Vincenzo Librandi, Aug 15 2013 *)
LinearRecurrence[{1,0,0,1,-1},{0,0,1,0,1},100] (* Harvey P. Dale, Feb 09 2015 *)
a[ n_] := Quotient[ n - 1, 2] - Quotient[ n, 4]; (* Michael Somos, May 05 2015 *)
  • PARI
    {a(n) = (n - 1) \ 2 - (n \ 4)}; /* Michael Somos, Oct 14 2008 */
  • PARI
    {a(n) = if( n<1, -a(3 - n), polcoeff( x^3 * (1 - x + x^2) / (1 - x - x^4 + x^5) + x * O(x^n), n))}; /* Michael Somos, Oct 14 2008 */

Formula

a(2*n + 2) = a(2*n - 1) = A004526(n).
a(n) = A005044(n) - A005044(n-6).
From Vladeta Jovovic, Dec 29 2001: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5).
G.f.: x^3*(1 - x + x^2)/(1 - x - x^4 + x^5). (End)
The g.f. can also be written as x^3 * (1 + x^3) / ((1 - x^2) * (1 - x^4)). - Michael Somos, May 05 2015
Euler transform of length 6 sequence [0, 1, 1, 1, 0, -1]. - Michael Somos, Oct 14 2008
a(n) = -a(3 - n) for all n in Z. - Michael Somos. Oct 14 2008
a(n) = abs(floor((n-1)*(-1)^n/4)). - Wesley Ivan Hurt, Oct 22 2013
a(n) = abs(A178804(n+1) - A178804(n)). - Reinhard Zumkeller, Nov 15 2014
a(n) = floor(n/2) - floor(n/4) - (1 if n even). - David Pasino, Jun 17 2016
E.g.f.: (4 - sin(x) - cos(x) + x*sinh(x) + (x - 3)*cosh(x))/4. - Ilya Gutkovskiy, Jun 21 2016
a(n) = floor((n-1)/2) - floor(n/4), n >= 0 (from the preceding a(n) formula). - Wolfdieter Lang, May 08 2017
a(n) = (2*n - 3 - 2*cos(n*Pi/2) - 3*cos(n*Pi) - 2*sin(n*Pi/2))/8. - Wesley Ivan Hurt, Oct 01 2017
a(n) = Sum_{i=1..floor((n-1)/2)} (n-i-1) mod 2. - Wesley Ivan Hurt, Nov 17 2017

Extensions

More terms from Sascha Kurz, Mar 25 2002

A008642 Quarter-squares repeated.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 6, 6, 9, 9, 12, 12, 16, 16, 20, 20, 25, 25, 30, 30, 36, 36, 42, 42, 49, 49, 56, 56, 64, 64, 72, 72, 81, 81, 90, 90, 100, 100, 110, 110, 121, 121, 132, 132, 144, 144, 156, 156, 169, 169, 182, 182, 196, 196, 210, 210, 225, 225
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The area of the largest rectangle whose perimeter is not greater than n. - Dmitry Kamenetsky, Aug 30 2006
Also number of partitions of n into parts 1, 2 or 4. - Reinhard Zumkeller, Aug 12 2011
Let us consider a rectangle composed of unit squares. Then count how many squares are necessary to surround this rectangle by a layer whose width is 1 unit. And repeat this surrounding ad libitum. This sequence, prepended by 4 zeros and with offset 0, gives the number of rectangles that need 2*n unit squares in one of their surrounding layers. - Michel Marcus, Sep 19 2015
a(n) is the number of nonnegative integer solutions (x,y,z) for n-2 <= 2*x + 3*y + 4*z <= n. For example, the two solutions for 1 <= 2*x + 3*y + 4*z <= 3 are (1,0,0) and (0,1,0). - Ran Pan, Oct 07 2015
Conjecture: Consider the number of compositions of n>=4*k+8 into odd parts, where the order of the parts 1,3,..,2k+1 does not count. Then, as k approaches infinity, a(n-4*k-8) is equal to the number of these restricted compositions minus A000009(n), the number of strict partitions of n. - Gregory L. Simay, Aug 12 2016
From Gus Wiseman, May 17 2019: (Start)
Also the number of length-3 integer partitions of n + 4 whose largest part is greater than the sum of the other two. These are unordered triples that cannot be the sides of a triangle. For example, the a(1) = 1 through a(10) = 9 partitions are (A = 10, B = 11, C = 12):
(311) (411) (421) (521) (522) (622) (632) (732) (733) (833)
(511) (611) (531) (631) (641) (741) (742) (842)
(621) (721) (722) (822) (751) (851)
(711) (811) (731) (831) (832) (932)
(821) (921) (841) (941)
(911) (A11) (922) (A22)
(931) (A31)
(A21) (B21)
(B11) (C11)
(End)
This sequence, prepended by four 0's and with offset 0, is the number of partitions of n into four parts whose smallest two parts are equal. - Wesley Ivan Hurt, Jan 05 2021
This sequence, prepended by four 0's and with offset 0, is the number of incongruent obtuse triangles formed from the vertices of a regular n-gon. - Frank M Jackson, Nov 27 2022

References

  • D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 105.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 112, D(n).

Links

Crossrefs

Cf. A002620.
Cf. A001399, A005044 (triangles without self-intersections), A069905, A124278, A266223, A325686, A325689, A325690, A325691, A325695.

Programs

  • Magma
    [Floor(((n+1)*((-1)^n+n+6)+9)/16): n in [0..70]]; // Vincenzo Librandi, Apr 02 2014
  • Maple
    seq((7/8+(-1)^k/8 + k + k^2/4)$2, k=0..100); # Robert Israel, Oct 08 2015
  • Mathematica
    CoefficientList[Series[1/((1-x)(1-x^2)(1-x^4)), {x, 0, 70}], x] (* Vincenzo Librandi, Apr 02 2014 *)
LinearRecurrence[{1,1,-1,1,-1,-1,1},{1,1,2,2,4,4,6}, 70] (* Harvey P. Dale, Jun 03 2015 *)
Table[Floor[((n + 1) ((-1)^n + n + 6) + 9)/16], {n, 0, 70}] (* Michael De Vlieger, Aug 14 2016 *)
  • PARI
    Vec(1/((1-x)*(1-x^2)*(1-x^4)) + O(x^70)) \\ Michel Marcus, Mar 31 2014
  • PARI
    vector(70, n, n--; floor(((n+1)*((-1)^n+n+6)+9)/16)) \\ Altug Alkan, Oct 08 2015
  • Sage
    [floor(floor(n/2+2)^2/2)/2 for n in (0..70)] # Bruno Berselli, Mar 03 2016

Formula

G.f.: 1/((1-x)*(1-x^2)*(1-x^4)).
a(n) = (2*n^2 + 14*n + 21 + (2*n + 7)*(-1)^n)/32 + ((1 + (-1)^n)/2 - (1 - (-1)^n)*i/2)*i^n/8, with i = sqrt(-1).
a(n) = floor(((n+1)*((-1)^n+n+6)+9)/16). - Tani Akinari, Jun 16 2013
a(n) = Sum_{i=1..floor((n+6)/2)} floor((n+6-2*i-(n mod 2))/4). - Wesley Ivan Hurt, Mar 31 2014
a(0)=1, a(1)=1, a(2)=2, a(3)=2, a(4)=4, a(5)=4, a(6)=6; for n>6, a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-4) - a(n-5) - a(n-6) + a(n-7). - Harvey P. Dale, Jun 03 2015
a(n) = floor(floor(n/2+2)^2/4) = floor(floor(n/2+2)^2/2)/2. - Bruno Berselli, Mar 03 2016
E.g.f.: ((14 + 7*x + x^2)*cosh(x) + 2*(cos(x) + sin(x)) + (7 + 9*x + x^2)*sinh(x))/16. - Stefano Spezia, Mar 05 2023
a(n) = floor((n + 4)/4)*floor((n + 6)/4). - Ridouane Oudra, Apr 01 2023
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