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%I #58 Aug 01 2022 16:43:01
%S 0,0,3,5,12,16,27,33,48,56,75,85,108,120,147,161,192,208,243,261,300,
%T 320,363,385,432,456,507,533,588,616,675,705,768,800,867,901,972,1008,
%U 1083,1121,1200,1240,1323,1365,1452,1496,1587,1633,1728,1776,1875,1925
%N a(n) = n^2 - floor((n+1)/2)^2.
%C The answer to a question from Mike and Laurie Crain (2crains(AT)concentric.net): how many even numbers are there in an n X n multiplication table starting at 1 X 1?
%C a(n+1) is the number of pairs (x,y) with x and y in {0,...,n}, x and y of the same parity, and x+y >= n. - _Clark Kimberling_, Jul 02 2012
%C From _J. M. Bergot_, Aug 08 2013: (Start)
%C Define a triangle to have T(1,1)=0 and T(n,c) = n^2 - c^2. The difference of the sum of the terms in antidiagonal(n+1) and those in antidiagonal(n)=a(n).
%C Column 0 is vertical and T(n,n)=0. The first few rows are 0; 3,0; 8,5,0; 15,12,7,0; 24,21,16,9,0; 35,32,27,20,11,0; the first few antidiagonals are 0; 3; 8,0; 15,5; 24,12,0; 35,21,7; 48,32,16,0; the first few sum of terms in the antidiagonals are 0, 3, 8, 20, 36, 63, 96, 144, 200, 275, 360, 468, 588, 735, 896, 1088, 1296, 1539. (End)
%C Sum of the largest parts in the partitions of 2n into two distinct odd parts. For example, a(5) = 16; the partitions of 2(5) = 10 into two distinct odd parts are (9,1) and (7,3). The sum of the largest parts is then 9+7 = 16. - _Wesley Ivan Hurt_, Nov 27 2017
%H Vincenzo Librandi, <a href="/A032438/b032438.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F a(n) = n^2 - A008794(n+1).
%F G.f.: x^2*(x^2 + 2*x + 3)/(1-x^2)^2/(1-x). - _Ralf Stephan_, Jun 10 2003
%F a(n) = (1/8)*(2*n*(3*n-1)+(2*n+1)*(-1)^n-1). a(-n-1) = A014255(n). - _Bruno Berselli_, Sep 27 2011
%F a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5), n > 4. - _Harvey P. Dale_, Nov 24 2011
%F E.g.f.: (x*(1 + 3*x)*cosh(x) + (3*x^2 + 3*x - 1)*sinh(x))/4. - _Stefano Spezia_, Aug 01 2022
%p A032438:=n->n^2-floor((n+1)/2)^2; seq(A032438(n), n=0..100) # _Wesley Ivan Hurt_, Nov 25 2013
%t Table[n^2-Floor[((n+1)/2)]^2,{n,0,50}] (* or *) LinearRecurrence[ {1,2,-2,-1,1},{0,0,3,5,12},51]
%o (Magma) [n^2-Floor( (n+1)/2 )^2 : n in [0..60]]; // _Vincenzo Librandi_, Sep 27 2011
%o (PARI) a(n)=n^2 - ((n+1)\2)^2 \\ _Charles R Greathouse IV_, Feb 19 2017
%Y First differences are in A059029, partial sums in A143785.
%Y Cf. A008794, A014255.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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