cp's OEIS Frontend

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

Showing 1-10 of 10 results.

A172131 Partial sums of floor(n^2/9) (A056838).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 8, 13, 20, 29, 40, 53, 69, 87, 108, 133, 161, 193, 229, 269, 313, 362, 415, 473, 537, 606, 681, 762, 849, 942, 1042, 1148, 1261, 1382, 1510, 1646, 1790, 1942, 2102, 2271, 2448, 2634, 2830, 3035, 3250, 3475, 3710, 3955, 4211, 4477, 4754
Offset: 0

Views

Author

Mircea Merca, Nov 19 2010

Keywords

Examples

			a(6) = 8 = 0 + 0 + 0 + 1 + 1 + 2 + 4.

Links

Crossrefs

Cf. A056838.

Programs

  • Magma
    [Round((2*n^3+3*n^2-15*n-9)/54): n in [0..60]]; // Vincenzo Librandi, Jun 25 2011
  • Maple
    a:= n-> round((2*n^3+3*n^2-15*n-9)/54): seq (a(n), n=0..50);
  • Mathematica
    Accumulate[Floor[Range[0,50]^2/9]] (* or *) LinearRecurrence[{3,-3,1,0,0,0,0,0,1,-3,3,-1},{0,0,0,1,2,4,8,13,20,29,40,53},60] (* Harvey P. Dale, Jan 10 2020 *)

Formula

a(n) = Sum_{k=0..n} floor(k^2/9).
a(n) = round((2*n^3 + 3*n^2 - 15*n - 9)/54).
a(n) = round((2*n^3 + 3*n^2 - 15*n - 8)/54).
a(n) = floor((2*n^3 + 3*n^2 - 15*n + 18)/54).
a(n) = ceiling((2*n^3 + 3*n^2 - 15*n - 34)/54).
a(n) = a(n-9) + (n-4)^2 + 4, n > 8.
G.f.: x^3*(x+1)*(x^2 - x + 1)^2/((x-1)^4*(x^2 + x + 1)*(x^6 + x^3 + 1)). [Colin Barker, Oct 26 2012]

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

Views

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

A007590 a(n) = floor(n^2/2).

Original entry on oeis.org

0, 0, 2, 4, 8, 12, 18, 24, 32, 40, 50, 60, 72, 84, 98, 112, 128, 144, 162, 180, 200, 220, 242, 264, 288, 312, 338, 364, 392, 420, 450, 480, 512, 544, 578, 612, 648, 684, 722, 760, 800, 840, 882, 924, 968, 1012, 1058, 1104, 1152, 1200, 1250, 1300, 1352, 1404
Offset: 0

Views

Author

N. J. A. Sloane, R. K. Guy

Keywords

Comments

Arithmetic mean of a pair of successive triangular numbers. - Amarnath Murthy, Jul 24 2005
Maximum sum of absolute differences of cyclically adjacent elements in a permutation of (1..n). For example, with n = 9, permutation (1,9,2,8,3,7,4,6,5) has adjacent differences (8,7,6,5,4,3,2,1,4) with maximal sum a(9) = 40. - Joshua Zucker, Dec 15 2005
a(n) = maximum number of non-overlapping 1 X 2 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's tool, see links. - Dmitry Kamenetsky, Aug 03 2009 [This is easily provable - David W. Wilson, Jan 25 2014]
Number of strictly increasing arrangements of 3 nonzero numbers in -(n+1)..(n+1) with sum zero. For example, a(2) = 2 has two solutions: (-3, 1, 2) and (-2, -1, 3) each add to zero. - Michael Somos, Apr 11 2011
For n >= 4 is a(n) the minimal value v such that v = Sum_{i in S1} i = Product_{j in S2} j with disjoint union of S1, S2 = {1, 2, ..., n+1}. Example: a(4) = 8 = 3+5 = 1*2*4. - Claudio Meller, May 27 2012
Sum_{n > 1} 1/a(n) = (zeta(2) + 1)/2. - Enrique Pérez Herrero, Jun 19 2013
Apart from the initial term this is the elliptic troublemaker sequence R_n(2,4) 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 12 2013
Maximum sum of displacements of elements in a permutation of (1..n). For example, with n = 9, permutation (5,6,7,8,9,1,2,3,4) has displacements (4,4,4,4,4,5,5,5,5) with maximal sum a(9) = 40. - David W. Wilson, Jan 25 2014
A245575(a(n)) mod 2 = 1, or for n > 0, where odd terms occur in A245575. - Reinhard Zumkeller, Aug 05 2014
Also the matching number of the n X n king, rook, and rook complement graphs. - Eric W. Weisstein, Jun 20 and Sep 14 2017
For n > 1, also the vertex count of the n X n white bishop graph. - Eric W. Weisstein, Jun 27 2017
This is also the number of distinct ways n^2 can be represented as the sum of two positive integers. - William Boyles, Jan 15 2018
Also the crossing number of the complete bipartite graph K_{4,n+1}. - Eric W. Weisstein, Sep 11 2018
The sequence can be obtained from A033429 by deleting the last digit of each term. - Bruno Berselli, Sep 11 2019
Starting at n=2, the number of facets of the n-dimensional Kunz cone C_(n+1). - Emily O'Sullivan, Jul 08 2023

Examples

			a(3) = 4 because 3^2/2 = 9/2 = 4.5 and floor(4.5) = 4.
a(4) = 8 because 4^2/2 = 16/2 = 8.
a(5) = 12 because 5^2/2 = 25/2 = 12.5 and floor(12.5) = 12.

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Column 3 of triangle A094953.
For n > 2: a(n) = sum of (n-1)-th row in triangle A101037.
Cf. A000212, A000290, A056827, A118013, A118015.
A080476 is essentially the same sequence.
Cf. A000982.
Elliptic troublemaker sequences: A000212 (= R_n(1,3) = R_n(2,3)), A002620 (= R_n(1,2)), 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. A182834 (complement), A245575.
First differences: A052928(n+1), is first differences of A212964; partial sums: A212964(n+1), is partial sums of A052928. - Guenther Schrack, Dec 10 2017
Cf. A033429 (5*n^2).
Cf. A056834, A130519, A056838, A056865.

Programs

  • Haskell
    a007550 = flip div 2 . (^ 2)  -- Reinhard Zumkeller, Aug 05 2014
  • Haskell
    a007590 = 0 : 0 : 0 : [ a1 + a2 - a3 + 2 | (a1, a2, a3) <- zip3 (tail (tail a007590)) (tail a007590) a007590 ] -- Luc Duponcheel, Sep 30 2020
  • Magma
    [Floor(n^2/2): n in [0..53]]; // Bruno Berselli, Mar 28 2011
  • Magma
    [Binomial(n,2)+Floor(n/2): n in [0..60]]; // Bruno Berselli, Jun 08 2017
  • Maple
    A007590:=n->floor(n^2/2); seq(A007590(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013
  • Mathematica
    Floor[Range[0, 53]^2/2] (* Alonso del Arte, Aug 07 2013 *)
Table[Binomial[n, 2] + Floor[n/2], {n, 0, 60}] (* Bruno Berselli, Jun 08 2017 *)
LinearRecurrence[{2, 0, -2, 1}, {0, 2, 4, 8}, 20] (* Eric W. Weisstein, Sep 14 2017 *)
CoefficientList[Series[-2 x/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* Eric W. Weisstein, Sep 14 2017 *)
Table[Floor[n^2/2], {n, 0, 20}] (* Eric W. Weisstein, Sep 11 2018 *)
  • PARI
    {a(n) = n^2 \ 2}
  • PARI
    {a(n) = local(v, c, m); m = n+1; forvec( v = vector( 3, i, [-m, m]), if( 0==prod( k=1, 3, v[k]), next); if( 0==sum( k=1, 3, v[k]), c++), 2); c} /* Michael Somos, Apr 11 2011 */
  • PARI
    first(n) = Vec(2*x^2/((1+x)*(1-x)^3) + O(x^n), -n); \\ Iain Fox, Dec 11 2017
  • Python
    def A007590(n): return n**2//2 # Chai Wah Wu, Jun 07 2022

Formula

a(n) = a(n-1) + a(n-2) - a(n-3) + 2 = 2*A002620(n) = A000217(n+1) + A004526(n). - Henry Bottomley, Mar 08 2000
a(n+1) = Sum_{k=1..n} (k + (k mod 2)). Therefore a(n) = Sum_{k=1..n} 2*floor(k/2). - William A. Tedeschi, Mar 19 2008
From R. J. Mathar, Nov 22 2008: (Start)
G.f.: 2*x^2/((1+x)*(1-x)^3).
a(n+1) - a(n) = A052928(n+1). (End)
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - R. H. Hardin, Mar 28 2011
a(n) = (2*n^2 + (-1)^n - 1)/4. - Bruno Berselli, Mar 28 2011
a(n) = ceiling((n^2-1)/2) = binomial(n+1, 2) - ceiling(n/2). - Wesley Ivan Hurt, Mar 08 2014, Jun 14 2013
a(n+1) = A014105(n) - A032528(n). - Richard R. Forberg, Aug 07 2013
a(n) = binomial(n,2) + floor(n/2). - Bruno Berselli, Jun 08 2017
a(n) = A099392(n+1) - 1. - Guenther Schrack, Dec 10 2017
E.g.f.: (x*(x + 1)*cosh(x) + (x^2 + x - 1)*sinh(x))/2. - Stefano Spezia, May 06 2021
From Amiram Eldar, Mar 20 2022: (Start)
Sum_{n>=2} 1/a(n) = Pi^2/12 + 1/2.
Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 - 1/2. (End)

Extensions

Edited by Charles R Greathouse IV, Apr 20 2010

A000212 a(n) = floor(n^2/3).

Original entry on oeis.org

0, 0, 1, 3, 5, 8, 12, 16, 21, 27, 33, 40, 48, 56, 65, 75, 85, 96, 108, 120, 133, 147, 161, 176, 192, 208, 225, 243, 261, 280, 300, 320, 341, 363, 385, 408, 432, 456, 481, 507, 533, 560, 588, 616, 645, 675, 705, 736, 768, 800, 833, 867, 901, 936
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Let M_n be the n X n matrix of the following form: [3 2 1 0 0 0 0 0 0 0 / 2 3 2 1 0 0 0 0 0 0 / 1 2 3 2 1 0 0 0 0 0 / 0 1 2 3 2 1 0 0 0 0 / 0 0 1 2 3 2 1 0 0 0 / 0 0 0 1 2 3 2 1 0 0 / 0 0 0 0 1 2 3 2 1 0 / 0 0 0 0 0 1 2 3 2 1 / 0 0 0 0 0 0 1 2 3 2 / 0 0 0 0 0 0 0 1 2 3]. Then for n > 2 a(n) = det M_(n-2). - Benoit Cloitre, Jun 20 2002
Largest possible size for the directed Cayley graph on two generators having diameter n - 2. - Ralf Stephan, Apr 27 2003
It seems that for n >= 2, a(n) is the maximum number of non-overlapping 1 X 3 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's tool, see links. - Dmitry Kamenetsky, Aug 03 2009
Maximum number of edges in a K4-free graph with n vertices. - Yi Yang, May 23 2012
3a(n) + 1 = y^2 if n is not 0 mod 3 and 3a(n) = y^2 otherwise. - Jon Perry, Sep 10 2012
Apart from the initial term this is the elliptic troublemaker sequence R_n(1, 3) (also sequence R_n(2, 3)) 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
The number of partitions of 2n into exactly 3 parts. - Colin Barker, Mar 22 2015
a(n-1) is the maximum number of non-overlapping triples (i,k), (i+1, k+1), (i+2, k+2) in an n X n matrix. Details: The triples are distributed along the main diagonal and 2*(n-1) other diagonals. Their maximum number is floor(n/3) + 2*Sum_{k = 1..n-1} floor(k/3) = floor((n-1)^2/3). - Gerhard Kirchner, Feb 04 2017
Conjecture: a(n) is the number of intersection points of n cevians that cut a triangle into the maximum number of pieces (see A007980). - Anton Zakharov, May 07 2017
From Gus Wiseman, Oct 05 2020: (Start)
Also the number of unimodal triples (meaning the middle part is not strictly less than both of the other two) of positive integers summing to n + 1. The a(2) = 1 through a(6) = 12 triples are:
(1,1,1) (1,1,2) (1,1,3) (1,1,4) (1,1,5)
(1,2,1) (1,2,2) (1,2,3) (1,2,4)
(2,1,1) (1,3,1) (1,3,2) (1,3,3)
(2,2,1) (1,4,1) (1,4,2)
(3,1,1) (2,2,2) (1,5,1)
(2,3,1) (2,2,3)
(3,2,1) (2,3,2)
(4,1,1) (2,4,1)
(3,2,2)
(3,3,1)
(4,2,1)
(5,1,1)
(End)

Examples

			G.f. = x^2 + 3*x^3 + 5*x^4 + 8*x^5 + 12*x^6 + 16*x^7 + 21*x^8 + 27*x^9 + 33*x^10 + ...
From _Gus Wiseman_, Oct 07 2020: (Start)
The a(2) = 1 through a(6) = 12 partitions of 2*n into exactly 3 parts (Barker) are the following. The Heinz numbers of these partitions are given by the intersection of A014612 (triples) and A300061 (even sum).
  (2,1,1)  (2,2,2)  (3,3,2)  (4,3,3)  (4,4,4)
           (3,2,1)  (4,2,2)  (4,4,2)  (5,4,3)
           (4,1,1)  (4,3,1)  (5,3,2)  (5,5,2)
                    (5,2,1)  (5,4,1)  (6,3,3)
                    (6,1,1)  (6,2,2)  (6,4,2)
                             (6,3,1)  (6,5,1)
                             (7,2,1)  (7,3,2)
                             (8,1,1)  (7,4,1)
                                      (8,2,2)
                                      (8,3,1)
                                      (9,2,1)
                                      (10,1,1)
(End)

References

  • 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. A000290, A007590 (= R_n(2,4)), A002620 (= R_n(1,2)), A118015, A056827, A118013.
Cf. 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)), A033440, A033441, A033442, A033443, A033444.
Cf. A001353 and A004523 (first differences). A184535 (= R_n(2,5) = R_n(3,5)).
Cf. A238738. - Bruno Berselli, Apr 17 2015
Cf. A007980
Cf. A005408.
A000217(n-2) counts 3-part compositions.
A014612 ranks 3-part partitions, with strict case A007304.
A069905 counts the 3-part partitions.
A211540 counts strict 3-part partitions.
A337453 ranks strict 3-part compositions.
Cf. A056834, A130519, A056838, A056865.
A001399(n-6)*4 is the strict version.
A001523 counts unimodal compositions, with strict case A072706.
A001840(n-4) is the non-unimodal version.
A001399(n-6)*2 is the strict non-unimodal version.
A007052 counts unimodal patterns.
A115981 counts non-unimodal compositions, ranked by A335373.
A011782 counts unimodal permutations.
A335373 is the complement of a ranking sequence for unimodal compositions.
A337459 ranks these compositions, with complement A337460.
Cf. A069905, A220377, A332743, A337461, A337561, A337603, A337604.

Programs

  • Magma
    [Floor(n^2 / 3): n in [0..50]]; // Vincenzo Librandi, May 08 2011
  • Maple
    A000212:=(-1+z-2*z**2+z**3-2*z**4+z**5)/(z**2+z+1)/(z-1)**3; # Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading 1.
A000212 := proc(n) option remember; `if`(n<4, [0,0,1,3][n+1], a(n-1)+a(n-3) -a(n-4)+2) end; # Peter Luschny, Nov 20 2011
  • Mathematica
    Table[Quotient[n^2, 3], {n, 0, 59}] (* Michael Somos, Jan 22 2014 *)
  • PARI
    {a(n) = n^2 \ 3}; /* Michael Somos, Sep 25 2006 */
  • Python
    def A000212(n): return n**2//3 # Chai Wah Wu, Jun 07 2022

Formula

G.f.: x^2*(1+x)/((1-x)^2*(1-x^3)). - Franklin T. Adams-Watters, Apr 01 2002
Euler transform of length 3 sequence [ 3, -1, 1]. - Michael Somos, Sep 25 2006
G.f.: x^2 * (1 - x^2) / ((1 - x)^3 * (1 - x^3)). a(-n) = a(n). - Michael Somos, Sep 25 2006
a(n) = Sum_{k = 0..n} A011655(k)*(n-k). - Reinhard Zumkeller, Nov 30 2009
a(n) = a(n-1) + a(n-3) - a(n-4) + 2 for n >= 4. - Alexander Burstein, Nov 20 2011
a(n) = a(n-3) + A005408(n-2) for n >= 3. - Alexander Burstein, Feb 15 2013
a(n) = (n-1)^2 - a(n-1) - a(n-2) for n >= 2. - Richard R. Forberg, Jun 05 2013
Sum_{n >= 2} 1/a(n) = (27 + 6*sqrt(3)*Pi + 2*Pi^2)/36. - Enrique Pérez Herrero, Jun 29 2013
0 = a(n)*(a(n+2) + a(n+3)) + a(n+1)*(-2*a(n+2) - a(n+3) + a(n+4)) + a(n+2)*(a(n+2) - 2*a(n+3) + a(n+4)) for all n in Z. - Michael Somos, Jan 22 2014
a(n) = Sum_{k = 1..n} k^2*A049347(n+2-k). - Mircea Merca, Feb 04 2014
a(n) = Sum_{i = 1..n+1} (ceiling(i/3) + floor(i/3) - 1). - Wesley Ivan Hurt, Jun 06 2014
a(n) = Sum_{j = 1..n} Sum_{i=1..n} ceiling((i+j-n-1)/3). - Wesley Ivan Hurt, Mar 12 2015
a(n) = Sum_{i = 1..n} floor(2*i/3). - Wesley Ivan Hurt, May 22 2017
a(-n) = a(n). - Paul Curtz, Jan 19 2020
a(n) = A001399(2*n - 3). - Gus Wiseman, Oct 07 2020
a(n) = (1/6)*(2*n^2 - 3 + gcd(n,3)). - Ridouane Oudra, Apr 15 2021
E.g.f.: (exp(x)*(-2 + 3*x*(1 + x)) + 2*exp(-x/2)*cos(sqrt(3)*x/2))/9. - Stefano Spezia, Oct 24 2022
Sum_{n>=2} (-1)^n/a(n) = Pi/sqrt(3) - Pi^2/36 - 3/4. - Amiram Eldar, Dec 02 2022

Extensions

Edited by Charles R Greathouse IV, Apr 19 2010

A130519 a(n) = Sum_{k=0..n} floor(k/4). (Partial sums of A002265.)

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 112, 120, 128, 136, 144, 153, 162, 171, 180, 190, 200, 210, 220, 231, 242, 253, 264, 276, 288, 300, 312, 325, 338, 351, 364, 378, 392, 406, 420, 435, 450
Offset: 0

Views

Author

Hieronymus Fischer, Jun 01 2007

Keywords

Comments

Complementary to A130482 with respect to triangular numbers, in that A130482(n) + 4*a(n) = n(n+1)/2 = A000217(n).
Disregarding the first three 0's the resulting sequence a'(n) is the sum of the positive integers <= n that have the same residue modulo 4 as n. This is the additive counterpart of the quadruple factorial numbers. - Peter Luschny, Jul 06 2011
From Heinrich Ludwig, Dec 23 2017: (Start)
Column sums of (shift of rows = 4):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
1 2 3 4 5 6 7 8 9 10 ...
1 2 3 4 5 6 ...
1 2 ...
.......................................
---------------------------------------
1 2 3 4 6 8 10 12 15 18 21 24 28 32 ...
shift of rows = 1 see A000217
shift of rows = 2 see A002620
shift of rows = 3 see A001840
shift of rows = 5 see A130520
(End)
Conjecture: a(n+2) is the maximum effective weight of a numerical semigroup S of genus n (see Nathan Pflueger). - Stefano Spezia, Jan 04 2019

Examples

			G.f. = x^4 + 2*x^5 + 3*x^6 + 4*x^7 + 6*x^8 + 8*x^9 + 10*x^10 + 12*x^11 + ...
[ n] a(n)
---------
[ 4] 1
[ 5] 2
[ 6] 3
[ 7] 4
[ 8] 1 + 5
[ 9] 2 + 6
[10] 3 + 7
[11] 4 + 8

Links

Crossrefs

Cf. A000217, A001840, A002264, A002265, A002266, A002620, A004526, A010872, A010873, A010874, A130481, A130483, A130520.
Cf. A000290, A007590, A000212, A118015, A056827, A056834, A056838, A056865.

Programs

  • GAP
    a:=List([0..65],n->Sum([0..n],k->Int(k/4)));; Print(a); # Muniru A Asiru, Jan 04 2019
  • Magma
    [Round(n*(n-2)/8): n in [0..70]]; // Vincenzo Librandi, Jun 25 2011
  • Maple
    quadsum := n -> add(k, k = select(k -> k mod 4 = n mod 4, [$1 .. n])):
A130519 := n ->`if`(n<3,0,quadsum(n-3)); seq(A130519(n),n=0..58); # Peter Luschny, Jul 06 2011
  • Mathematica
    a[ n_] := Quotient[ (n - 1)^2, 8]; (* Michael Somos, Oct 14 2011 *)
  • Maxima
    makelist(floor((n-1)^2/8), n, 0, 70); /* Stefano Spezia, Jan 04 2019 */
  • PARI
    {a(n) = (n - 1)^2 \ 8}; /* Michael Somos, Oct 14 2011 */
  • Python
    def A130519(n): return (n-1)**2>>3  # Chai Wah Wu, Jul 30 2022

Formula

G.f.: x^4/((1-x^4)*(1-x)^2) = x^4/((1+x)*(1+x^2)*(1-x)^3).
a(n) = +2*a(n-1) -1*a(n-2) +1*a(n-4) -2*a(n-5) +1*a(n-6).
a(n) = floor(n/4)*(n - 1 - 2*floor(n/4)) = A002265(n)*(n - 1 - 2*A002265(n)).
a(n) = (1/2)*A002265(n)*(n - 2 + A010873(n)).
a(n) = floor((n-1)^2/8). - Mitch Harris, Sep 08 2008
a(n) = round(n*(n-2)/8) = round((n^2-2*n-1)/8) = ceiling((n+1)*(n-3)/8). - Mircea Merca, Nov 28 2010
a(n) = A001972(n-4), n>3. - Franklin T. Adams-Watters, Jul 10 2009
a(n) = a(n-4)+n-3, n>3. - Mircea Merca, Nov 28 2010
Euler transform of length 4 sequence [ 2, 0, 0, 1]. - Michael Somos, Oct 14 2011
a(n) = a(2-n) for all n in Z. - Michael Somos, Oct 14 2011
a(n) = A214734(n, 1, 4). - Renzo Benedetti, Aug 27 2012
a(4n) = A000384(n), a(4n+1) = A001105(n), a(4n+2) = A014105(n), a(4n+3) = A046092(n). - Philippe Deléham, Mar 26 2013
a(n) = Sum_{i=1..ceiling(n/2)-1} (i mod 2) * (n - 2*i - 1). - Wesley Ivan Hurt, Jan 23 2014
a(n) = ( 2*n^2-4*n-1+(-1)^n+2*((-1)^((2*n-1+(-1)^n)/4)-(-1)^((6*n-1+(-1)^n)/4)) )/16 = ( 2*n*(n-2) - (1-(-1)^n)*(1-2*i^(n*(n-1))) )/16, where i=sqrt(-1). - Luce ETIENNE, Aug 29 2014
E.g.f.: (1/8)*((- 1 + x)*x*cosh(x) + 2*sin(x) + (- 1 - x + x^2)*sinh(x)). - Stefano Spezia, Jan 15 2019
a(n) = (A002620(n-1) - A011765(n+1)) / 2, for n > 0. - Yuchun Ji, Feb 05 2021
Sum_{n>=4} 1/a(n) = Pi^2/12 + 5/2. - Amiram Eldar, Aug 13 2022

Extensions

Partially edited by R. J. Mathar, Jul 11 2009

A056827 a(n) = floor(n^2/6).

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 28, 32, 37, 42, 48, 54, 60, 66, 73, 80, 88, 96, 104, 112, 121, 130, 140, 150, 160, 170, 181, 192, 204, 216, 228, 240, 253, 266, 280, 294, 308, 322, 337, 352, 368, 384, 400, 416, 433, 450, 468, 486, 504
Offset: 0

Views

Author

N. J. A. Sloane, Sep 02 2000

Keywords

Comments

a(n-1) represents the floor of the area under the polygon connecting the lattice points (n, floor(n/3)) from 0..n, n>0 (see example). - Wesley Ivan Hurt, Jun 06 2014

Examples

			5|                                             .__.__.
.|                                            /|  |  |
4|                                    .__.__./_|__|__|
.|                                   /|  |  |  |  |  |
3|                           .__.__./_|__|__|__|__|__|
.|                          /|  |  |  |  |  |  |  |  |
2|                  .__.__./_|__|__|__|__|__|__|__|__|
.|                 /|  |  |  |  |  |  |  |  |  |  |  |
1|         .__.__./_|__|__|__|__|__|__|__|__|__|__|__|
.|        /|  |  |  |  |  |  |  |  |  |  |  |  |  |  |
0|.__.__./_|__|__|__|__|__|__|__|__|__|__|__|__|__|__|_________________
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17   .. n
  0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42 48   .. a(n)
     0  0  0  1  2  4  6  8 10 13 16 20 24 28 32 37 42   .. a(n-1) <--

Links

Crossrefs

Cf. A000290, A007590, A000212, A002620, A118015, A118013, A056834, A130519, A056838, A056865.

Programs

Formula

From R. J. Mathar, Nov 22 2008: (Start)
G.f.: x^3*(1+x^2)/((1+x)*(1-x)^3*(1+x+x^2)*(1-x+x^2)).
a(n+1) - a(n) = A123919(n). (End)
a(n) = floor( (1/2) * Sum_{i=1..n+1} (ceiling(i/3) + floor(i/3) - 1) ). - Wesley Ivan Hurt, Jun 06 2014
Sum_{n>=3} 1/a(n) = 15/8 + Pi^2/36 - Pi/(4*sqrt(3)) + tan(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)). - Amiram Eldar, Aug 13 2022

A118015 a(n) = floor(n^2/5).

Original entry on oeis.org

0, 0, 0, 1, 3, 5, 7, 9, 12, 16, 20, 24, 28, 33, 39, 45, 51, 57, 64, 72, 80, 88, 96, 105, 115, 125, 135, 145, 156, 168, 180, 192, 204, 217, 231, 245, 259, 273, 288, 304, 320, 336, 352, 369, 387, 405, 423, 441, 460, 480, 500, 520, 540, 561, 583, 605, 627, 649, 672
Offset: 0

Views

Author

Reinhard Zumkeller, Apr 10 2006

Keywords

Comments

It seems that for n >= 5, a(n) is the maximum number of non-overlapping 1 X 5 rectangles that can be packed into an n X n square. Rectangles can only be placed parallel to the sides of the square. Verified with Lobato's program. - Dmitry Kamenetsky, Aug 03 2009
Ismailescu & Lee prove that for n > 6, a(n) is composite. - Charles R Greathouse IV, Jan 10 2025

Links

Crossrefs

Cf. A000212, A000290, A007590, A002620, A056827, A118013, A279169.
Cf. A056834, A130519, A056838, A056865.

Programs

Formula

G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4)*(1 - x)^3). - Klaus Brockhaus, Nov 18 2008
a(n) = A008732(n-4) + A008732(n-3). - R. J. Mathar, Nov 22 2008
a(5*m+r) = m*(5*m + 2*r) + a(r), with m >= 0 and 0 <= r < 5. Example: for m=4 and r=3, a(5*4+3) = a(23) = 4*(5*4 + 2*3) + a(3) = 104 + 1 = 105. - Bruno Berselli, Dec 12 2016
Sum_{n>=3} 1/a(n) = 25/16 + Pi^2/30 + sqrt(5-2*sqrt(5))*Pi/4. - Amiram Eldar, Aug 13 2022

Extensions

Edited by Charles R Greathouse IV, Apr 20 2010

A056865 a(n) = floor(n^2/10).

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 57, 62, 67, 72, 78, 84, 90, 96, 102, 108, 115, 122, 129, 136, 144, 152, 160, 168, 176, 184, 193, 202, 211, 220, 230, 240, 250, 260, 270, 280, 291, 302, 313, 324
Offset: 0

Views

Author

N. J. A. Sloane, Sep 02 2000

Keywords

Links

Crossrefs

Cf. A056864.
Cf. A000290, A007590, A000212, A002620, A118015, A056827, A056834, A130519, A056838.

Programs

  • Maple
    A056865 := proc(n)
    floor(n^2/10) ;
end proc:
seq(A056865(n),n=0..100) ; # R. J. Mathar, Mar 08 2016
  • Mathematica
    Floor[Range[0,60]^2/10] (* or *) LinearRecurrence[{2,-1,0,0,0,0,0,0,0,1,-2,1},{0,0,0,0,1,2,3,4,6,8,10,12},60] (* Harvey P. Dale, Jun 29 2022 *)
  • PARI
    a(n) = n^2\10; \\ Michel Marcus, Mar 08 2016

Formula

G.f.: -x^4*(1+x^4) / ( (1+x)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)*(x-1)^3 ). - R. J. Mathar, Mar 08 2016

A056834 a(n) = floor(n^2/7).

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 24, 28, 32, 36, 41, 46, 51, 57, 63, 69, 75, 82, 89, 96, 104, 112, 120, 128, 137, 146, 155, 165, 175, 185, 195, 206, 217, 228, 240, 252, 264, 276, 289, 302, 315, 329, 343, 357, 371, 386, 401, 416, 432, 448
Offset: 0

Views

Author

N. J. A. Sloane, Sep 02 2000

Keywords

Links

Crossrefs

Cf. A000290, A007590, A000212, A002620, A118015, A056827, A130519, A056838, A056865.

Programs

  • Mathematica
    Floor[(Range[0,60]^2)/7] (* or *) LinearRecurrence[{2,-1,0,0,0,0,1,-2,1},{0,0,0,1,2,3,5,7,9},60] (* Harvey P. Dale, Jul 21 2014 *)
CoefficientList[Series[-x^3 (1 + x) (x^2 - x + 1)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1) (x - 1)^3), {x, 0, 100}], x] (* Vincenzo Librandi, Jul 22 2014 *)
  • PARI
    a(n) = n^2\7; \\ Michel Marcus, Mar 03 2022

Formula

a(n) = +2*a(n-1) -a(n-2) +a(n-7) -2*a(n-8) +a(n-9).
G.f.: -x^3*(1+x)*(x^2-x+1) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^3 ).

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6
Offset: 1

Views

Author

Jason Kimberley, Nov 09 2016

Keywords

Examples

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Links

Crossrefs

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

Programs

  • Magma
    A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];
  • Mathematica
    Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

Formula

T(n,k) = A010766(n^2,k).
Showing 1-10 of 10 results.

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