A332610 Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

4, 14, 48, 32, 102, 128, 70, 192, 204, 288, 124, 326, 312, 396, 512, 226, 524, 516, 600, 716, 928, 360, 802, 784, 868, 984, 1196, 1472, 566, 1192, 1196, 1280, 1396, 1608, 1884, 2304, 820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328, 1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928

Offset

1,1

Comments

See A331457 for illustrations.

Links

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

Formula

The first column is A324042, for which there is an explicit formula.

No formula is known for column 2, which is A332606.

For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(m^2+n^2)+12*n+4*m-24 + 4*V(m,m,2)+4*V(n,n,2), 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:
[4],
[14, 48],
[32, 102, 128],
[70, 192, 204, 288],
[124, 326, 312, 396, 512],
[226, 524, 516, 600, 716, 928],
[360, 802, 784, 868, 984, 1196, 1472],
[566, 1192, 1196, 1280, 1396, 1608, 1884, 2304],
[820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328],
[1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928],
[1696, 3074, 3456, 3540, 3656, 3868, 4144, 4564, 5080, 5884, 6848],
[2310, 4052, 4684, 4768, 4884, 5096, 5372, 5792, 6308, 7112, 8076, 9312],
...

Crossrefs

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

Sequence in context: A000908 A121095 A264816 * A015651 A022632 A027906

Adjacent sequences: A332607 A332608 A332609 * A332611 A332612 A332613

Keyword

nonn,tabl

Author

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

Status

approved