A271917 Array read by antidiagonals: T(m,n) (m>=1, n>=1) = f(m,n) if m <= n or f(n,m) if n < m, where f(m,n) = m*(m^2-1)*(2*n-m)/12.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 14, 20, 14, 5, 0, 0, 6, 18, 30, 30, 18, 6, 0, 0, 7, 22, 40, 50, 40, 22, 7, 0, 0, 8, 26, 50, 70, 70, 50, 26, 8, 0, 0, 9, 30, 60, 90, 105, 90, 60, 30, 9, 0, 0, 10, 34, 70, 110, 140, 140, 110, 70, 34, 10, 0

Offset: 1

Author

N. J. A. Sloane, Apr 26 2016

Keywords

• nonn
• tabl

Comments

T(m,n) is the number of ways to choose four distinct points from an m X n rectangular grid that form a square. See A271916 for the version where we count only squares aligned with the axes.

Examples

			The array begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
0, 2, 6, 10, 14, 18, 22, 26, 30, 34, ...
0, 3, 10, 20, 30, 40, 50, 60, 70, 80, ...
0, 4, 14, 30, 50, 70, 90, 110, 130, 150, ...
0, 5, 18, 40, 70, 105, 140, 175, 210, 245, ...
0, 6, 22, 50, 90, 140, 196, 252, 308, 364, ...
0, 7, 26, 60, 110, 175, 252, 336, 420, 504, ...
0, 8, 30, 70, 130, 210, 308, 420, 540, 660, ...
0, 9, 34, 80, 150, 245, 364, 504, 660, 825, ...
...
As a triangle:
0,
0, 0,
0, 1, 0,
0, 2, 2, 0,
0, 3, 6, 3, 0,
0, 4, 10, 10, 4, 0,
0, 5, 14, 20, 14, 5, 0,
0, 6, 18, 30, 30, 18, 6, 0,
0, 7, 22, 40, 50, 40, 22, 7, 0,
0, 8, 26, 50, 70, 70, 50, 26, 8, 0,
...
		

Crossrefs

Main diagonal is A002415 (shifted).

Cf. A227133, A271916.

Programs

Maple

f1:=(m,n)->(1/12)*m*(m^2-1)*(2*n-m);
f2:=(m,n)->if n>=m then f1(m,n) else f1(n,m) fi;
for m from 1 to 10 do
lprint([seq(f2(m,n),n=1..10)]); od;