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A332611 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).

%I A332611 #17 Jul 20 2020 13:09:03
%S A332611 0,2,8,14,36,80,34,92,144,208,90,194,280,356,504,154,336,432,520,680,
%T A332611 856,288,554,724,824,996,1184,1512,462,812,1096,1208,1392,1592,1932,
%U A332611 2352,742,1314,1680,1804,2000,2212,2564,2996,3640,1038,1756,2296,2432,2640,2864,3228,3672,4328,5016
%N A332611 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 A332611 See A331457 for illustrations.
%F A332611 The first column is A324043, for which there is an explicit formula.
%F A332611 No formula is known for column 2, which is A332607.
%F A332611 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 A332611 Triangle begins:
%e A332611 [0],
%e A332611 [2, 8],
%e A332611 [14, 36, 80],
%e A332611 [34, 92, 144, 208],
%e A332611 [90, 194, 280, 356, 504],
%e A332611 [154, 336, 432, 520, 680, 856],
%e A332611 [288, 554, 724, 824, 996, 1184, 1512],
%e A332611 [462, 812, 1096, 1208, 1392, 1592, 1932, 2352],
%e A332611 [742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],
%e A332611 [1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],
%e A332611 [1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],
%e A332611 [2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],
%e A332611 ....
%Y A332611 Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.
%K A332611 nonn,tabl
%O A332611 1,2
%A A332611 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 12 2020