A332599 - OEIS

%I #25 Mar 13 2020 12:40:46

%S 5,13,37,35,99,152,75,213,256,364,159,401,448,568,776,275,657,704,836,

%T 1056,1340,477,1085,1132,1276,1508,1804,2272,755,1619,1712,1868,2112,

%U 2420,2900,3532,1163,2327,2552,2720,2976,3296,3788,4432,5336,1659,3257,3568,3748,4016,4348,4852,5508,6424,7516

%N 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 See A331457 and A331776 for further illustrations.

%C 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 Scott R. Shannon, <a href="/A331776/a331776.png">Colored illustration for T(3,3) = 152</a>

%H Scott R. Shannon, <a href="/A331776/a331776_1.png">Colored illustration for T(4,4) = 364</a>

%H N. J. A. Sloane, <a href="/A331457/a331457.pdf">Illustration for T(3,3) = 152.</a>

%F Column 1 is A331755, for which there is an explicit formula.

%F Column 2 is A331763, for which no formula is known.

%F 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 Triangle begins:

%e [5],

%e [13, 37],

%e [35, 99, 152],

%e [75, 213, 256, 364],

%e [159, 401, 448, 568, 776],

%e [275, 657, 704, 836, 1056, 1340],

%e [477, 1085, 1132, 1276, 1508, 1804, 2272],

%e [755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],

%e [1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],

%e [1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],

%e ...

%Y Cf. A331457, A331755, A331763, A331776, A332599, A332600.

%Y The main diagonal is A332598.

%K nonn,tabl

%O 1,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Mar 03 2020

%E More terms from _N. J. A. Sloane_, Mar 13 2020