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%I #26 Feb 29 2020 23:25:16
%S 0,1,10,44,130,313,640,1192,2044,3305,5078,7524,10750,14993,20388,
%T 27128,35448,45665,57922,72636,89970,110297,133976,161440,192860,
%U 228857,269758,316012,367974,426417,491468,564120,644640,733633,831674,939292
%N The number of rectangles (orthogonal or not) with corners on an n X n grid of points.
%H David Radcliffe, <a href="/A085582/b085582.txt">Table of n, a(n) for n = 1..1000</a>
%H David Radcliffe, <a href="/A085582/a085582.py.txt">Python script to calculate a(n)</a>
%F a(n) = A000537(n-1) + A113751(n). - _T. D. Noe_, Nov 09 2005 [corrected by _David Radcliffe_, Feb 06 2020]
%F a(n) = n*(n-1)^2*(2n-1)/6 + 2*Sum_{a,b>0, 0<s<r<n, gcd(r,s)=1} max(n-a*s-b*r,0)*max(n-a*r-b*s,0). - _David Radcliffe_, Feb 06 2020
%e a(3) = 10 because on the 3 X 3 grid there are four 1 X 1 rectangles, two 1 X 2s, two 2 X 1's, one 2 X 2 and one 45-degree rectangle, sqrt(2) X sqrt(2).
%Y Cf. A000537, A002415, A113751 (diagonal rectangles on an n X n grid).
%K nonn
%O 1,3
%A Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 06 2003
%E Edited by _Don Reble_, Nov 05 2005
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