A332610 - OEIS

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

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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],