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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
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
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
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved