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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). 2
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified January 19 04:56 EST 2022. Contains 350464 sequences. (Running on oeis4.)