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

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

Author

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

Keywords

• nonn
• easy

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

• 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).

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