2,3

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.

a(n)=A085582(n)-A000537(n)

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.

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, 2, 42}]

Cf. A000537 (parallel rectangles on an n X n grid), A085582 (all rectangles on an n X n grid).

Adjacent sequences: A113748 A113749 A113750 this_sequence A113752 A113753 A113754

Sequence in context: A063489 A002417 A126858 this_sequence A107233 A098213 A050477

nonn

T. D. Noe (noe(AT)sspectra.com), Nov 09 2005