cp's OEIS Frontend

A340688 Irregular table read by rows: Take a concave circular triangle with all diagonals drawn, as in A340685. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.

%I A340688 #15 Mar 07 2021 20:56:46
%S A340688 1,12,22,3,3,66,36,67,108,12,222,186,48,6,265,465,132,6,582,786,174,
%T A340688 48,732,1905,324,76,3,6,1410,2268,558,156,6,1704,3732,861,223,18,3,
%U A340688 2778,4242,1260,324,42,3369,6540,1872,409,42,24,4896,7302,2502,540,72,24,6138,10467,3306,907,99,30
%N A340688 Irregular table read by rows: Take a concave circular triangle with all diagonals drawn, as in A340685. Then T(n,k) = number of k-sided polygons in that figure for k >= 3.
%C A340688 See A340685 for images of the regions and A340686 for images of the vertices.
%H A340688 Scott R. Shannon, <a href="/A340688/a340688.png">Image of the regions for n = 20</a>.
%H A340688 Wikipedia, <a href="https://en.wikipedia.org/wiki/Circular_triangle">Circular triangle</a>.
%e A340688 A concave circular triangle with 1 point dividing its edges, n = 2, contains 12 triangles and no other n-gons, so the second row is [12]. A concave circular triangle with 2 points dividing its edges, n = 3, contains 22 triangles, 3 quadrilaterals, 3 pentagons and no other n-gons, so the third row is [22, 3, 3].
%e A340688 The table begins:
%e A340688 1;
%e A340688 12;
%e A340688 22, 3, 3;
%e A340688 66, 36;
%e A340688 67, 108, 12;
%e A340688 222, 186, 48, 6;
%e A340688 265, 465, 132, 6;
%e A340688 582, 786, 174, 48;
%e A340688 732, 1905, 324, 76, 3, 6;
%e A340688 1410, 2268, 558, 156, 6;
%e A340688 1704, 3732, 861, 223, 18, 3;
%e A340688 2778, 4242, 1260, 324, 42;
%e A340688 3369, 6540, 1872, 409, 42, 24;
%e A340688 4896, 7302, 2502, 540, 72, 24;
%e A340688 6138, 10467, 3306, 907, 99, 30;
%e A340688 8364, 12522, 4566, 1020, 120, 18;
%e A340688 10132, 16149, 5439, 1410, 288, 57, 0, 3;
%e A340688 13398, 19308, 6870, 1962, 252, 30, 12;
%e A340688 16029, 23082, 8859, 2422, 336, 90, 3;
%e A340688 20682, 29658, 10800, 2976, 528, 66;
%Y A340688 Cf. A340685 (regions), A340686 (vertices), A340687 (edges), A340614, A007678, A092867.
%K A340688 nonn,tabf
%O A340688 1,2
%A A340688 _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 16 2021