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%I #155 Dec 27 2023 11:21:06
%S 0,0,1,1,4,4,9,9,16,16,25,25,36,36,49,49,64,64,81,81,100,100,121,121,
%T 144,144,169,169,196,196,225,225,256,256,289,289,324,324,361,361,400,
%U 400,441,441,484,484,529,529,576,576
%N Squares repeated; a(n) = floor(n/2)^2.
%C Also number of non-attacking kings on (n-1) X (n-1) board (cf. A030978). - Koksal Karakus (karakusk(AT)hotmail.com), May 27 2002
%C Also the independence number and clique covering number of the (n-1) X (n-1) king graph. - _Eric W. Weisstein_, Jun 20 2017
%C Maximum number of 2 X 2 tiles that fit on an n X n board. - _Jon Perry_, Aug 10 2003
%C (n)-(1) + (n-1)-(2) + (n-3)-(3) + ... + (n-r)-(r) ... n terms. E.g., 5-1+4-2+3 = 9, 6-1+5-2+4-3 = 9, 7-1+6-2+5-3+4 = 16, 8-1+7-2+6-3+5-4 = 16. - _Amarnath Murthy_, Jul 24 2005
%C The smallest possible number of white cells in a solution to an n X n nurikabe grid. - _Tanya Khovanova_, Feb 24 2009
%C (1 + x + 4*x^2 + 4*x^3 + 9*x^4 + ...) = (1/(1-x))*(1 + 3*x^2 + 5*x^4 + 7*x^6 + ...). - _Gary W. Adamson_, Apr 07 2010
%C If the set {1,2,...,n} is divided in half (a part having size ceiling(n/2) and the rest), then a(n+1) is the largest possible difference between the totals of these parts. - _Vladimir Shevelev_, Oct 14 2017
%C a(n+1) is the sum of the smallest parts of the partitions of 2n into two odd parts. - _Wesley Ivan Hurt_, Dec 06 2017
%C a(n-1) is the largest number of single cells of an n X n grid that share no edge or vertex with each other or those of the grid perimeter. - _Stefano Spezia_, Jul 30 2021
%C The binomial transform is 0, 0, 1, 4, 14, 44, 128, 352, 928, 2368, 5888... (see A007466). - _R. J. Mathar_, Feb 25 2023
%H Vincenzo Librandi, <a href="/A008794/b008794.txt">Table of n, a(n) for n = 0..10000</a>
%H Stefano Spezia, <a href="/A008794/a008794.jpg">Illustration of initial terms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CliqueCoveringNumber.html">Clique Covering 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/KingsProblem.html">Kings Problem</a>.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F G.f.: x^2*(1 + x^2)/((1 - x)*(1 - x^2)^2).
%F a(n) = floor(n/2)^2.
%F From _Paul Barry_, May 31 2003: (Start)
%F a(n) = (2*n - 1)*(-1)^n/8 + (2*n^2 - 2*n + 1)/8.
%F a(n+1) = Sum_{k=0..n} k*(1-(-1)^k)/2. (End)
%F a(n+2) = Sum_{k=0..n} A109613(k)*A059841(n-k). - _Reinhard Zumkeller_, Dec 05 2009
%F a(n) = A182579(n,n-2) for n > 1. - _Reinhard Zumkeller_, May 07 2012
%F 3*a(n) = A032766(n)^2 - A032766(n^2). - _Bruno Berselli_, Oct 21 2016
%F a(n) = Sum_{i=1..n-1; i odd} i. - _Olivier Pirson_, Nov 06 2017
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - _Iain Fox_, Dec 08 2017
%F E.g.f.: ((x^2 - x)*cosh(x) + (1 + x + x^2)*sinh(x))/4. - _Stefano Spezia_, Oct 07 2018
%p A008794:=n->floor(n/2)^2: seq(A008794(n), n=0..50); # _Wesley Ivan Hurt_, Dec 08 2017
%t With[{sq = Range[0, 30]^2}, Riffle[sq, sq]] (* _Harvey P. Dale_, Nov 20 2015 *)
%t Table[Floor[n/2]^2, {n, 0, 49}] (* _Michael De Vlieger_, Oct 21 2016 *)
%t Table[(2 n - 1) (-1)^n/8 + (2 n^2 - 2 n + 1)/8, {n, 0, 49}] (* _Michael De Vlieger_, Oct 21 2016 *)
%t CoefficientList[Series[x^2*(1 + x^2)/((1 - x) (1 - x^2)^2), {x, 0, 49}], x] (* _Michael De Vlieger_, Oct 21 2016 *)
%t CoefficientList[Series[((x^2-x)Cosh[x]+(1+x+x^2)Sinh[x])/4,{x,0,50}],x]*Table[k!,{k,0,50}] (* _Stefano Spezia_, Oct 07 2018 *)
%o (Magma) [(2*n-1)*(-1)^n/8+(2*n^2-2*n +1)/8: n in [0..60]]; // _Vincenzo Librandi_, Aug 21 2011
%o (PARI) a(n)=(n\2)^2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (PARI) first(n) = Vec(x^2*(1 + x^2)/((1 - x)*(1 - x^2)^2) + O(x^n), -n) \\ _Iain Fox_, Dec 08 2017
%o (GAP) Flat(List([0..24],n->[n^2,n^2])); # _Muniru A Asiru_, Oct 09 2018
%o (Sage) [((-1)^n*(2*n-1) +(2*n^2-2*n +1))/8 for n in (0..50)] # _G. C. Greubel_, Sep 11 2019
%o (Python)
%o def A008794(n): return (n//2)**2 # _Chai Wah Wu_, Jun 07 2022
%Y Cf. A030978, A032766, A059841, A086832, A109613, A182579, A189889.
%K nonn,easy
%O 0,5
%A _N. J. A. Sloane_
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