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

1,2

Comments

See A331457 for illustrations.

Links

Table of n, a(n) for n=1..55.

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).

Example

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.

Sequence in context: A046959 A086177 A299337 * A162484 A333575 A259119

Adjacent sequences: A332608 A332609 A332610 * A332612 A332613 A332614

Keyword

nonn,tabl

Author

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

Status

approved