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%I #71 Nov 13 2023 07:29:26
%S 1,13,4,82,67,11,307,406,206,24,841,1441,1216,489,50,1891,3796,4211,
%T 2835,995,80,3718,8299,10901,9672,5671,1802,154,6637,15982,23536,
%U 24780,19139,10196,3052,220,11017,28081,44906,53109,48686,34166,17011,4810,375
%N Table read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the n*(k+1) boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives number of regions in the resulting planar graph.
%C "In general position" implies that the internal lines (or chords) formed from the n*k edge points only have simple intersections; there is no interior points where three or more such chords meet. Note that for even-n n-gons, with n>=6, the chords from the n corner points do create non-simple intersections.
%C Note that although the number of regions with a given number of edges in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices created from the edge-point chords remain simple.
%H Scott R. Shannon, <a href="/A366253/a366253.png">Image for T(5,3)</a>.
%H Scott R. Shannon, <a href="/A366253/a366253_1.png">Image for T(6,2)</a>.
%H Scott R. Shannon, <a href="/A366253/a366253_2.png">Image for T(8,2)</a>.
%H Scott R. Shannon, <a href="/A366253/a366253_3.png">Image for T(10,2)</a>.
%F T(n,k) = A367190(n,k) - A367183(n,k) + 1 by Euler's formula.
%F Conjectured:
%F T(3,k) = A367118(k) = (9/4)*k^4 + 3*k^3 + (15/4)*k^2 + 3*k + 1.
%F T(4,k) = A367121(k) = (17/2)*k^4 + 19*k^3 + (43/2)*k^2 + 14*k + 4.
%F T(5,k) = (45/2)*k^4 + 60*k^3 + 70*k^2 + (85/2)*k + 11.
%F T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (687/4)*k^2 + 102*k + 24.
%F T(7,k) = (371/4)*k^4 + 287*k^3 + (1421/4)*k^2 + 210*k + 50.
%F T(8,k) = 161*k^4 + 518*k^3 + 655*k^2 + 388*k + 80.
%F T(9,k) = 261*k^4 + 864*k^3 + (2223/2)*k^2 + (1323/2)*k + 154.
%F T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (7085/4)*k^2 + 1060*k + 220.
%e The table begins:
%e 1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417,...
%e 4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497,...
%e 11, 206, 1216, 4211, 10901, 23536, 44906, 78341, 127711, 197426, 292436,...
%e 24, 489, 2835, 9672, 24780, 53109, 100779, 175080, 284472, 438585, 648219,...
%e 50, 995, 5671, 19139, 48686, 103825, 196295, 340061, 551314, 848471, 1252175,...
%e 80, 1802, 10196, 34166, 86480, 183770, 346532, 599126, 969776, 1490570,...
%e 154, 3052, 17011, 56611, 142696, 302374, 569017, 982261, 1588006, 2438416,...
%e 220, 4810, 26705, 88495, 222400, 470270, 883585, 1523455, 2460620, 3775450,...
%e 375, 7305, 40096, 132243, 331431, 699535, 1312620, 2260941, 3648943, 5595261,...
%e 444, 10509, 57810, 190263, 475980, 1003269, 1880634, 3236775, 5220588, 8001165,...
%e 781, 14938, 81082, 265747, 663391, 1396396, 2615068, 4497637, 7250257,...
%e 952, 20335, 110439, 361354, 900844, 1894347, 3544975, 6093514, 9818424,...
%e 1456, 27391, 147421, 480931, 1197076, 2514781, 4702741, 8079421, 13013056,...
%e 1696, 35716, 192552, 627484, 1560352, 3275556, 6122056, 10513372,...
%e .
%e .
%e .
%Y Cf. A367118 (first row), A367121 (second row), A007678 (first column), A367183 (vertices), A367190 (edges).
%K nonn,tabl
%O 3,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 09 2023