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%I M1090 #196 Jul 10 2023 10:39:57
%S 0,0,2,4,8,12,18,24,32,40,50,60,72,84,98,112,128,144,162,180,200,220,
%T 242,264,288,312,338,364,392,420,450,480,512,544,578,612,648,684,722,
%U 760,800,840,882,924,968,1012,1058,1104,1152,1200,1250,1300,1352,1404
%N a(n) = floor(n^2/2).
%C Arithmetic mean of a pair of successive triangular numbers. - _Amarnath Murthy_, Jul 24 2005
%C 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
%C 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]
%C 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
%C 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
%C Sum_{n > 1} 1/a(n) = (zeta(2) + 1)/2. - _Enrique Pérez Herrero_, Jun 19 2013
%C 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
%C 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
%C A245575(a(n)) mod 2 = 1, or for n > 0, where odd terms occur in A245575. - _Reinhard Zumkeller_, Aug 05 2014
%C Also the matching number of the n X n king, rook, and rook complement graphs. - _Eric W. Weisstein_, Jun 20 and Sep 14 2017
%C For n > 1, also the vertex count of the n X n white bishop graph. - _Eric W. Weisstein_, Jun 27 2017
%C 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
%C Also the crossing number of the complete bipartite graph K_{4,n+1}. - _Eric W. Weisstein_, Sep 11 2018
%C The sequence can be obtained from A033429 by deleting the last digit of each term. - _Bruno Berselli_, Sep 11 2019
%C Starting at n=2, the number of facets of the n-dimensional Kunz cone C_(n+1). - _Emily O'Sullivan_, Jul 08 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007590/b007590.txt">Table of n, a(n) for n = 0..1000</a>
%H Laurent Bulteau, Samuele Giraudo and Stéphane Vialette, <a href="http://igm.univ-mlv.fr/~giraudo/Data/Papers/Disorders%20and%20permutations.pdf">Disorders and permutations </a>, 32nd Annual Symposium on Combinatorial Pattern Matching (CPM 2021). Article No. 18; pp. 18:1-18:14.
%H Bakir Farhi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Farhi/farhi7.html">On the Representation of the Natural Numbers as the Sum of Three Terms of the Sequence floor(n^2/a)</a>, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.4.
%H Richard K. Guy, <a href="/A002186/a002186.pdf">Letters to N. J. A. Sloane, June-August 1968</a>.
%H R. D. Lobato, <a href="https://web.archive.org/web/20210506153254/http://lagrange.ime.usp.br/~lobato/packing/run/">Recursive partitioning approach for the Manufacturer's Pallet Loading Problem</a>.
%H Emily O'Sullivan, <a href="/A007590/a007590.pdf">Understanding the face structure of the Kunz cone</a>, Master's thesis, San Diego State Univ., 2023.
%H K. E. Stange, <a href="http://arxiv.org/abs/1108.3051">Integral points on elliptic curves and explicit valuations of division polynomials</a> arXiv:1108.3051 [math.NT], 2011-2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphCrossingNumber.html">Graph Crossing Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KingGraph.html">King Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookComplementGraph.html">Rook Complement Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RookGraph.html">Rook Graph</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexCount.html">Vertex Count</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/WhiteBishopGraph.html">White Bishop Graph</a>.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-2,1).
%F 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
%F 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
%F From _R. J. Mathar_, Nov 22 2008: (Start)
%F G.f.: 2*x^2/((1+x)*(1-x)^3).
%F a(n+1) - a(n) = A052928(n+1). (End)
%F a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4). - _R. H. Hardin_, Mar 28 2011
%F a(n) = (2*n^2 + (-1)^n - 1)/4. - _Bruno Berselli_, Mar 28 2011
%F a(n) = ceiling((n^2-1)/2) = binomial(n+1, 2) - ceiling(n/2). - _Wesley Ivan Hurt_, Mar 08 2014, Jun 14 2013
%F a(n+1) = A014105(n) - A032528(n). - _Richard R. Forberg_, Aug 07 2013
%F a(n) = binomial(n,2) + floor(n/2). - _Bruno Berselli_, Jun 08 2017
%F a(n) = A099392(n+1) - 1. - _Guenther Schrack_, Dec 10 2017
%F E.g.f.: (x*(x + 1)*cosh(x) + (x^2 + x - 1)*sinh(x))/2. - _Stefano Spezia_, May 06 2021
%F From _Amiram Eldar_, Mar 20 2022: (Start)
%F Sum_{n>=2} 1/a(n) = Pi^2/12 + 1/2.
%F Sum_{n>=2} (-1)^n/a(n) = Pi^2/12 - 1/2. (End)
%e a(3) = 4 because 3^2/2 = 9/2 = 4.5 and floor(4.5) = 4.
%e a(4) = 8 because 4^2/2 = 16/2 = 8.
%e a(5) = 12 because 5^2/2 = 25/2 = 12.5 and floor(12.5) = 12.
%p A007590:=n->floor(n^2/2); seq(A007590(k), k=0..100); # _Wesley Ivan Hurt_, Oct 29 2013
%t Floor[Range[0, 53]^2/2] (* _Alonso del Arte_, Aug 07 2013 *)
%t Table[Binomial[n, 2] + Floor[n/2], {n, 0, 60}] (* _Bruno Berselli_, Jun 08 2017 *)
%t LinearRecurrence[{2, 0, -2, 1}, {0, 2, 4, 8}, 20] (* _Eric W. Weisstein_, Sep 14 2017 *)
%t CoefficientList[Series[-2 x/((-1 + x)^3 (1 + x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Sep 14 2017 *)
%t Table[Floor[n^2/2], {n, 0, 20}] (* _Eric W. Weisstein_, Sep 11 2018 *)
%o (PARI) {a(n) = n^2 \ 2}
%o (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 */
%o (PARI) first(n) = Vec(2*x^2/((1+x)*(1-x)^3) + O(x^n), -n); \\ _Iain Fox_, Dec 11 2017
%o (Magma) [Floor(n^2/2): n in [0..53]]; // _Bruno Berselli_, Mar 28 2011
%o (Magma) [Binomial(n,2)+Floor(n/2): n in [0..60]]; // _Bruno Berselli_, Jun 08 2017
%o (Haskell)
%o a007550 = flip div 2 . (^ 2) -- _Reinhard Zumkeller_, Aug 05 2014
%o (Haskell)
%o a007590 = 0 : 0 : 0 : [ a1 + a2 - a3 + 2 | (a1, a2, a3) <- zip3 (tail (tail a007590)) (tail a007590) a007590 ] -- _Luc Duponcheel_, Sep 30 2020
%o (Python)
%o def A007590(n): return n**2//2 # _Chai Wah Wu_, Jun 07 2022
%Y Column 3 of triangle A094953.
%Y For n > 2: a(n) = sum of (n-1)-th row in triangle A101037.
%Y Cf. A000212, A000290, A056827, A118013, A118015.
%Y A080476 is essentially the same sequence.
%Y Cf. A000982.
%Y 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)).
%Y Cf. A182834 (complement), A245575.
%Y 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
%Y Cf. A033429 (5*n^2).
%Y Cf. A056834, A130519, A056838, A056865.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, _R. K. Guy_
%E Edited by _Charles R Greathouse IV_, Apr 20 2010
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