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A113751
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Number of diagonal rectangles with corners on an n X n grid of points.
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6
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0, 0, 1, 8, 30, 88, 199, 408, 748, 1280, 2053, 3168, 4666, 6712, 9363, 12728, 16952, 22256, 28681, 36536, 45870, 56936, 69967, 85264, 102860, 123232, 146557, 173128, 203138, 237192, 275243, 318104, 365856, 418912, 477649, 542392, 613406, 691848
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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Jinyuan Wang, Table of n, a(n) for n = 1..1000
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FORMULA
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a(n) = A085582(n) - A000537(n-1). [corrected by David Radcliffe, Feb 06 2020]
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EXAMPLE
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a(3) = 1 because for the 3 X 3 grid, there is only one diagonal rectangle - a square having sides sqrt(2) units.
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.
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MATHEMATICA
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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}]
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CROSSREFS
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Cf. A000537 (parallel rectangles on an n X n grid), A085582 (all rectangles on an n X n grid).
Sequence in context: A126858 A232772 A213776 * A343520 A107233 A098213
Adjacent sequences: A113748 A113749 A113750 * A113752 A113753 A113754
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe, Nov 09 2005
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EXTENSIONS
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a(1) = 0 prepended by Jinyuan Wang, Feb 06 2020
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STATUS
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approved
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