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A007590 a(n) = floor(n^2/2).
(Formerly M1090)
64
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 (list; graph; refs; listen; history; text; internal format)
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 1x2 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 graph. - Eric W. Weisstein, Jun 20 2017

For n > 1, also the vertex count of the n X n white bishop graph. -  Eric W. Weisstein, Jun 27 2017

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

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

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

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, Matching Number

Eric Weisstein's World of Mathematics, King 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 + mod(k, 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

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

a[n_] := a[n] = 2a[n - 1] - 2a[n - 3] + a[n - 4]; a[0] = a[1] = 0; a[2] = 2; a[3] = 4; Array[a, 54, 0] (* Robert G. Wilson v, Mar 28 2011 *)

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 *)

PROG

(PARI) {a(n) = n^2 \ 2}

(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]]; //

(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 */

(Haskell)

a007550 = flip div 2 . (^ 2)  -- Reinhard Zumkeller, Aug 05 2014

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.

Sequence in context: A152125 A176562 A100057 * A080476 A256885 A053799

Adjacent sequences:  A007587 A007588 A007589 * A007591 A007592 A007593

KEYWORD

nonn,easy,changed

AUTHOR

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

EXTENSIONS

Edited by Charles R Greathouse IV, Apr 20 2010

STATUS

approved

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Last modified June 27 11:25 EDT 2017. Contains 288788 sequences.