A332611 - OEIS

%I #17 Jul 20 2020 13:09:03

%S 0,2,8,14,36,80,34,92,144,208,90,194,280,356,504,154,336,432,520,680,

%T 856,288,554,724,824,996,1184,1512,462,812,1096,1208,1392,1592,1932,

%U 2352,742,1314,1680,1804,2000,2212,2564,2996,3640,1038,1756,2296,2432,2640,2864,3228,3672,4328,5016

%N Triangle read by rows: T(m,n) = number of quadrilateral regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

%C See A331457 for illustrations.

%F The first column is A324043, for which there is an explicit formula.

%F No formula is known for column 2, which is A332607.

%F For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(3*m*n-m-4*n) + 2*(V(m,m,1)-2*V(m,m,2)-m^2-4*m+6) + 2*(V(n,n,1)-2*V(n,n,2)-n^2-4*n+6) where V(m,n,q) = Sum_{i = 1..m, j = 1..n, gcd(i,j)=q} (m+1-i)*(n+1-j).

%e Triangle begins:

%e [0],

%e [2, 8],

%e [14, 36, 80],

%e [34, 92, 144, 208],

%e [90, 194, 280, 356, 504],

%e [154, 336, 432, 520, 680, 856],

%e [288, 554, 724, 824, 996, 1184, 1512],

%e [462, 812, 1096, 1208, 1392, 1592, 1932, 2352],

%e [742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],

%e [1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],

%e [1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],

%e [2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],

%e ....

%Y Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.

%K nonn,tabl

%O 1,2

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 12 2020