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

3,1

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.

Links

Table of n, a(n) for n=3..47.

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).

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.

Example

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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Crossrefs

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

Sequence in context: A306364 A357819 A357821 * A214401 A009781 A266913

Adjacent sequences: A367180 A367181 A367182 * A367184 A367185 A367186

Keyword

nonn,tabl

Author

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

Status

approved