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A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

%I A332599 #25 Mar 13 2020 12:40:46
%S A332599 5,13,37,35,99,152,75,213,256,364,159,401,448,568,776,275,657,704,836,
%T A332599 1056,1340,477,1085,1132,1276,1508,1804,2272,755,1619,1712,1868,2112,
%U A332599 2420,2900,3532,1163,2327,2552,2720,2976,3296,3788,4432,5336,1659,3257,3568,3748,4016,4348,4852,5508,6424,7516
%N A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).
%C A332599 See A331457 and A331776 for further illustrations.
%C A332599 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 A332599 Scott R. Shannon, <a href="/A331776/a331776.png">Colored illustration for T(3,3) = 152</a>
%H A332599 Scott R. Shannon, <a href="/A331776/a331776_1.png">Colored illustration for T(4,4) = 364</a>
%H A332599 N. J. A. Sloane, <a href="/A331457/a331457.pdf">Illustration for T(3,3) = 152.</a>
%F A332599 Column 1 is A331755, for which there is an explicit formula.
%F A332599 Column 2 is A331763, for which no formula is known.
%F A332599 For m >= n >= 3, T(m,n) = A332600(m,n) - A331457(m,n) (Euler for genus 1 graph), and both A332600 and A331457 have explicit formulas.
%e A332599 Triangle begins:
%e A332599 [5],
%e A332599 [13, 37],
%e A332599 [35, 99, 152],
%e A332599 [75, 213, 256, 364],
%e A332599 [159, 401, 448, 568, 776],
%e A332599 [275, 657, 704, 836, 1056, 1340],
%e A332599 [477, 1085, 1132, 1276, 1508, 1804, 2272],
%e A332599 [755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],
%e A332599 [1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],
%e A332599 [1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],
%e A332599 ...
%Y A332599 Cf. A331457, A331755, A331763, A331776, A332599, A332600.
%Y A332599 The main diagonal is A332598.
%K A332599 nonn,tabl
%O A332599 1,1
%A A332599 _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 03 2020
%E A332599 More terms from _N. J. A. Sloane_, Mar 13 2020