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A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).

%I A332600 #19 Mar 13 2020 01:59:21
%S A332600 8,28,92,80,240,360,178,508,604,860,372,944,1040,1320,1792,654,1548,
%T A332600 1652,1956,2452,3124,1124,2520,2640,2968,3488,4184,5256,1782,3754,
%U A332600 4004,4356,4900,5620,6716,8188,2724,5392,5936,6312,6880,7624,8744,10240,12304,3914,7528,8364,8764,9356,10124,11268,12788,14876,17460
%N A332600 Triangle read by rows: T(n,k) = number of edges in a "frame" of size n X k (see Comments in A331457 for definition).
%C A332600 See A331457 and A331776 for further illustrations.
%C A332600 There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.
%H A332600 Scott R. Shannon, <a href="/A331776/a331776.png">Colored illustration for T(3,3) = 360</a>
%H A332600 Scott R. Shannon, <a href="/A331776/a331776_1.png">Colored illustration for T(4,4) = 860</a>
%H A332600 N. J. A. Sloane, <a href="/A331457/a331457.pdf">Illustration for T(3,3) = 360.</a>
%F A332600 Column 1 is A331757, for which there is an explicit formula.
%F A332600 Column 2 is A331765, for which no formula is known.
%F A332600 For m >= n >= 3, T(m,n) = (3*A332610(m,n)+4*A332611(m,n)+4*m+4*n-8)/2, and both A332610 and A332611 have explicit formulas.
%e A332600 Triangle begins:
%e A332600 [8],
%e A332600 [28, 92],
%e A332600 [80, 240, 360],
%e A332600 [178, 508, 604, 860],
%e A332600 [372, 944, 1040, 1320, 1792],
%e A332600 [654, 1548, 1652, 1956, 2452, 3124],
%e A332600 [1124, 2520, 2640, 2968, 3488, 4184, 5256],
%e A332600 [1782, 3754, 4004, 4356, 4900, 5620, 6716, 8188],
%e A332600 [2724, 5392, 5936, 6312, 6880, 7624, 8744, 10240, 12304],
%e A332600 [3914, 7528, 8364, 8764, 9356, 10124, 11268, 12788, 14876, 17460],
%e A332600 ...
%Y A332600 Cf. A331457, A331757, A331765, A331776, A332599, A332610, A332611.
%Y A332600 The main diagonal is A332597.
%K A332600 nonn,tabl
%O A332600 1,1
%A A332600 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 03 2020
%E A332600 More terms from _N. J. A. Sloane_, Mar 13 2020