cp's OEIS Frontend

A271916 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-1)*(3*n-m-1)/6.

%I A271916 #21 Aug 18 2024 09:49:24
%S A271916 0,0,0,0,1,0,0,2,2,0,0,3,5,3,0,0,4,8,8,4,0,0,5,11,14,11,5,0,0,6,14,20,
%T A271916 20,14,6,0,0,7,17,26,30,26,17,7,0,0,8,20,32,40,40,32,20,8,0,0,9,23,38,
%U A271916 50,55,50,38,23,9,0,0,10,26,44,60,70,70,60,44,26,10,0
%N A271916 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-1)*(3*n-m-1)/6.
%C A271916 T(m,n) is the number of ways to choose four distinct points from an m X n rectangular grid that form a square aligned with the axes. See A271917 for the count of all subsquares.
%e A271916 The array begins:
%e A271916 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A271916 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, ...
%e A271916 0, 2, 5, 8, 11, 14, 17, 20, 23, 26, ...
%e A271916 0, 3, 8, 14, 20, 26, 32, 38, 44, 50, ...
%e A271916 0, 4, 11, 20, 30, 40, 50, 60, 70, 80, ...
%e A271916 0, 5, 14, 26, 40, 55, 70, 85, 100, 115, ...
%e A271916 0, 6, 17, 32, 50, 70, 91, 112, 133, 154, ...
%e A271916 0, 7, 20, 38, 60, 85, 112, 140, 168, 196, ...
%e A271916 0, 8, 23, 44, 70, 100, 133, 168, 204, 240, ...
%e A271916 0, 9, 26, 50, 80, 115, 154, 196, 240, 285, ...
%e A271916 ...
%e A271916 As a triangle:
%e A271916 0,
%e A271916 0, 0,
%e A271916 0, 1, 0,
%e A271916 0, 2, 2, 0,
%e A271916 0, 3, 5, 3, 0,
%e A271916 0, 4, 8, 8, 4, 0,
%e A271916 0, 5, 11, 14, 11, 5, 0,
%e A271916 0, 6, 14, 20, 20, 14, 6, 0,
%e A271916 0, 7, 17, 26, 30, 26, 17, 7, 0,
%e A271916 0, 8, 20, 32, 40, 40, 32, 20, 8, 0,
%e A271916 ...
%p A271916 f1:=(m,n)->(1/6)*m*(m-1)*(3*n-m-1);
%p A271916 f2:=(m,n)->if n>=m then f1(m,n) else f1(n,m) fi;
%p A271916 for m from 1 to 10 do
%p A271916 lprint([seq(f2(m,n),n=1..10)]); od;
%Y A271916 See A115262 for another version.
%Y A271916 Main diagonal is A000330 (shifted).
%Y A271916 Cf. A227133, A271917.
%K A271916 nonn,tabl
%O A271916 1,8
%A A271916 _N. J. A. Sloane_, Apr 26 2016