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).

0, 2, 8, 14, 36, 80, 34, 92, 144, 208, 90, 194, 280, 356, 504, 154, 336, 432, 520, 680, 856, 288, 554, 724, 824, 996, 1184, 1512, 462, 812, 1096, 1208, 1392, 1592, 1932, 2352, 742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640, 1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

Keywords

• nonn
• tabl

Comments

See A331457 for illustrations.

Examples

			Triangle begins:
[0],
[2, 8],
[14, 36, 80],
[34, 92, 144, 208],
[90, 194, 280, 356, 504],
[154, 336, 432, 520, 680, 856],
[288, 554, 724, 824, 996, 1184, 1512],
[462, 812, 1096, 1208, 1392, 1592, 1932, 2352],
[742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],
[1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],
[1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],
[2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],
....
		

Crossrefs

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

Formula

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

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

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).