A367183 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 vertices in the resulting planar graph.

3, 12, 5, 72, 58, 10, 282, 375, 185, 19, 795, 1376, 1155, 451, 42, 1818, 3685, 4090, 2734, 938, 57, 3612, 8130, 10700, 9478, 5523, 1711, 135, 6492, 15743, 23235, 24463, 18858, 9981, 2943, 171, 10827, 27760, 44485, 52639, 48230, 33771, 16740, 4646, 341

Offset: 3

Author

Scott R. Shannon and N. J. A. Sloane, Nov 08 2023

Keywords

• nonn
• tabl

Comments

"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.

Examples

			The table begins:
3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987,...
5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728,...
10, 185, 1155, 4090, 10700, 23235, 44485, 77780, 126990, 196525, 291335,...
19, 451, 2734, 9478, 24463, 52639, 100126, 174214, 283363, 437203, 646534,...
42, 938, 5523, 18858, 48230, 103152, 195363, 338828, 549738, 846510, 1249787,...
57, 1711, 9981, 33771, 85849, 182847, 345261, 597451, 967641, 1487919, 2194237,...
135, 2943, 16740, 56106, 141885, 301185, 567378, 980100, 1585251, 2434995,...
171, 4646, 26336, 87831, 221351, 468746, 881496, 1520711, 2457131, 3771126,...
341, 7128, 39666, 131450, 330165, 697686, 1310078, 2257596, 3644685, 5589980,...
313, 10204, 57199, 189214, 474361, 1000948, 1877479, 3232654, 5215369, 7994716,...
728, 14677, 80457, 264602, 661570, 1393743, 2611427, 4492852, 7244172,...
771, 19909, 109586, 359892, 898591, 1891121, 3540594, 6087796, 9811187,...
1380, 27030, 146565, 479370, 1194600, 2511180, 4697805, 8072940, 13004820,...
1393, 35085, 191353, 625477, 1557297, 3271213, 6116185, 10505733,......
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Links

• Scott R. Shannon, Image for T(5,3).
• Scott R. Shannon, Image for T(6,2).
• Scott R. Shannon, Image for T(7,1).
• Scott R. Shannon, Image for T(8,1).

Crossrefs

Cf. A367117 (first row), A334698 (second row), A007569 (first column), A366253 (regions), A367190 (edges).

Formula

T(n,k) = A367190(n,k) - A366253(n,k) + 1 by Euler's formula.

T(3,k) = A367117(k) = (9/4)*k^4 + 3*k^3 + (3/4)*k^2 + 3*k + 3.

Conjectured:

T(4,k) = A334698(k+1) = (17/2)*k^4 + 19*k^3 + (31/2)*k^2 + 10*k + 5.

T(5,k) = (45/2)*k^4 + 60*k^3 + 60*k^2 + (65/2)*k + 10.

T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (627/4)*k^2 + 84*k + 19.

T(7,k) = (371/4)*k^4 + 287*k^3 + (1337/4)*k^2 + 182*k + 42.

T(8,k) = 161*k^4 + 518*k^3 + 627*k^2 + 348*k + 57.

T(9,k) = 261*k^4 + 864*k^3 + (2151/2)*k^2 + (1215/2)*k + 135.

T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (6905/4)*k^2 + 990*k + 171.