A007590 a(n) = floor(n^2/2). (Formerly M1090)

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

0,3

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

References

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, Disorders and permutations , 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.

Bakir Farhi, On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a), Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.

Richard K. Guy, Letters to N. J. A. Sloane, June-August 1968.

R. D. Lobato, Recursive partitioning approach for the Manufacturer's Pallet Loading Problem.

Emily O'Sullivan, Understanding the face structure of the Kunz cone, Master's thesis, San Diego State Univ., 2023.

K. E. Stange, Integral points on elliptic curves and explicit valuations of division polynomials arXiv:1108.3051 [math.NT], 2011-2014.

Eric Weisstein's World of Mathematics, Complete Bipartite Graph.

Eric Weisstein's World of Mathematics, Graph Crossing Number.

Eric Weisstein's World of Mathematics, Matching Number.

Eric Weisstein's World of Mathematics, King Graph.

Eric Weisstein's World of Mathematics, Rook Complement Graph.

Eric Weisstein's World of Mathematics, Rook Graph.

Eric Weisstein's World of Mathematics, Vertex Count.

Eric Weisstein's World of Mathematics, White Bishop Graph.

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

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)

Example

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.

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 *)

Prog

(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
(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
(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
(Python)
def A007590(n): return n**2//2 # Chai Wah Wu, Jun 07 2022

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.

Sequence in context: A176562 A337120 A100057 * A080476 A256885 A293495

Adjacent sequences: A007587 A007588 A007589 * A007591 A007592 A007593

Keyword

nonn,easy

Author

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

Extensions

Edited by Charles R Greathouse IV, Apr 20 2010

Status

approved