A271917 - OEIS

%I #17 Apr 26 2016 16:07:23

%S 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,

%T 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,

%U 60,90,105,90,60,30,9,0,0,10,34,70,110,140,140,110,70,34,10,0

%N 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.

%C 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.

%e The array begins:

%e 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

%e 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...

%e 0, 2, 6, 10, 14, 18, 22, 26, 30, 34, ...

%e 0, 3, 10, 20, 30, 40, 50, 60, 70, 80, ...

%e 0, 4, 14, 30, 50, 70, 90, 110, 130, 150, ...

%e 0, 5, 18, 40, 70, 105, 140, 175, 210, 245, ...

%e 0, 6, 22, 50, 90, 140, 196, 252, 308, 364, ...

%e 0, 7, 26, 60, 110, 175, 252, 336, 420, 504, ...

%e 0, 8, 30, 70, 130, 210, 308, 420, 540, 660, ...

%e 0, 9, 34, 80, 150, 245, 364, 504, 660, 825, ...

%e ...

%e As a triangle:

%e 0,

%e 0, 0,

%e 0, 1, 0,

%e 0, 2, 2, 0,

%e 0, 3, 6, 3, 0,

%e 0, 4, 10, 10, 4, 0,

%e 0, 5, 14, 20, 14, 5, 0,

%e 0, 6, 18, 30, 30, 18, 6, 0,

%e 0, 7, 22, 40, 50, 40, 22, 7, 0,

%e 0, 8, 26, 50, 70, 70, 50, 26, 8, 0,

%e ...

%p f1:=(m,n)->(1/12)*m*(m^2-1)*(2*n-m);

%p f2:=(m,n)->if n>=m then f1(m,n) else f1(n,m) fi;

%p for m from 1 to 10 do

%p lprint([seq(f2(m,n),n=1..10)]); od;

%Y Main diagonal is A002415 (shifted).

%Y Cf. A227133, A271916.

%K nonn,tabl

%O 1,8

%A _N. J. A. Sloane_, Apr 26 2016