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A113751 Number of diagonal rectangles with corners on an n X n grid of points.

%I A113751 #10 Feb 12 2020 08:27:14
%S A113751 0,0,1,8,30,88,199,408,748,1280,2053,3168,4666,6712,9363,12728,16952,
%T A113751 22256,28681,36536,45870,56936,69967,85264,102860,123232,146557,
%U A113751 173128,203138,237192,275243,318104,365856,418912,477649,542392,613406,691848
%N A113751 Number of diagonal rectangles with corners on an n X n grid of points.
%C A113751 The diagonal rectangles are the ones whose sides are not parallel to the grid axes. All the rectangles can be reflected so that the slope of one side is >= 1. There are a total of A046657(n-1) these slopes. These slopes are the basis of the Mathematica program that counts the rectangles.
%H A113751 Jinyuan Wang, <a href="/A113751/b113751.txt">Table of n, a(n) for n = 1..1000</a>
%F A113751 a(n) = A085582(n) - A000537(n-1). [corrected by _David Radcliffe_, Feb 06 2020]
%e A113751 a(3) = 1 because for the 3 X 3 grid, there is only one diagonal rectangle - a square having sides sqrt(2) units.
%e A113751 a(4) = 8 because for the 4 X 4 grid, there are 4 squares having sides sqrt(2) units, 2 squares having sides sqrt(5) units and 2 rectangles that are sqrt(2) by 2*sqrt(2) units.
%t A113751 Table[n=m-1; slopes=Union[Flatten[Table[a/b, {b, n}, {a, b, n-b}]]]; rects=0; Do[b=Numerator[slopes[[i]]]; a=Denominator[slopes[[i]]]; base={a+b, a+b}; l=0; While[l++; k=l; While[extent=base+{b, a}(k-1)+{a, b}(l-1); extent[[1]]<=n && extent[[2]]<=n, pos={n+1, n+1}-extent; If[a==b && k==l, fact=1, If[pos[[1]]==pos[[2]], fact=2, fact=4]]; rects=rects+fact*Times@@pos; k++ ]; k>l], {i, Length[slopes]}]; rects, {m, 1, 42}]
%Y A113751 Cf. A000537 (parallel rectangles on an n X n grid), A085582 (all rectangles on an n X n grid).
%K A113751 nonn
%O A113751 1,4
%A A113751 _T. D. Noe_, Nov 09 2005
%E A113751 a(1) = 0 prepended by _Jinyuan Wang_, Feb 06 2020