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

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

Offset

1,4

Comments

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.

Links

Jinyuan Wang, Table of n, a(n) for n = 1..1000

Formula

a(n) = A085582(n) - A000537(n-1). [corrected by David Radcliffe, Feb 06 2020]

Example

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.

Mathematica

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}]

Crossrefs

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

Keyword

nonn

Author

T. D. Noe, Nov 09 2005

Extensions

a(1) = 0 prepended by Jinyuan Wang, Feb 06 2020

Status

approved