cp's OEIS Frontend

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).

%I A332610 #19 Jul 20 2020 13:11:57
%S A332610 4,14,48,32,102,128,70,192,204,288,124,326,312,396,512,226,524,516,
%T A332610 600,716,928,360,802,784,868,984,1196,1472,566,1192,1196,1280,1396,
%U A332610 1608,1884,2304,820,1634,1704,1788,1904,2116,2392,2812,3328,1218,2296,2500,2584,2700,2912,3188,3608,4124,4928
%N 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).
%C A332610 See A331457 for illustrations.
%F A332610 The first column is A324042, for which there is an explicit formula.
%F A332610 No formula is known for column 2, which is A332606.
%F A332610 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).
%e A332610 Triangle begins:
%e A332610 [4],
%e A332610 [14, 48],
%e A332610 [32, 102, 128],
%e A332610 [70, 192, 204, 288],
%e A332610 [124, 326, 312, 396, 512],
%e A332610 [226, 524, 516, 600, 716, 928],
%e A332610 [360, 802, 784, 868, 984, 1196, 1472],
%e A332610 [566, 1192, 1196, 1280, 1396, 1608, 1884, 2304],
%e A332610 [820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328],
%e A332610 [1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928],
%e A332610 [1696, 3074, 3456, 3540, 3656, 3868, 4144, 4564, 5080, 5884, 6848],
%e A332610 [2310, 4052, 4684, 4768, 4884, 5096, 5372, 5792, 6308, 7112, 8076, 9312],
%e A332610 ...
%Y A332610 Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.
%K A332610 nonn,tabl
%O A332610 1,1
%A A332610 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 12 2020