A367190 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 edges in the resulting planar graph.

3, 24, 8, 153, 124, 20, 588, 780, 390, 42, 1635, 2816, 2370, 939, 91, 3708, 7480, 8300, 5568, 1932, 136, 7329, 16428, 21600, 19149, 11193, 3512, 288, 13128, 31724, 46770, 49242, 37996, 20176, 5994, 390, 21843, 55840, 89390, 105747, 96915, 67936, 33750, 9455, 715

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.

See A367183 and A366253 for images of the n-gons.

Links

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

Formula

T(n,k) = A367183(n,k) + A366253(n,k) - 1 by Euler's formula.

Conjectures:

T(3,k) = A367119(k) = (9/2)*k^4 + 6*k^3 + (9/2)*k^2 + 6*k + 3.

T(4,k) = A367122(k) = 17*k^4 + 38*k^3 + 37*k^2 + 24*k + 8.

T(5,k) = 45*k^4 + 120*k^3 + 130*k^2 + 75*k + 20.

T(6,k) = (195/2)*k^4 + 285*k^3 + (657/2)*k^2 + 186*k + 42.

T(7,k) = (371/2)*k^4 + 574*k^3 + (1379/2)*k^2 + 392*k + 91.

T(8,k) = 322*k^4 + 1036*k^3 + 1282*k^2 + 736*k + 136.

T(9,k) = 522*k^4 + 1728*k^3 + 2187*k^2 + 1269*k + 288.

T(10,k) = (1605/2)*k^4 + 2715*k^3 + (6995/2)*k^2 + 2050*k + 390.

Example

The table begins:
3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403,...
8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224,...
20, 390, 2370, 8300, 21600, 46770, 89390, 156120, 254700, 393950, 583770,...
42, 939, 5568, 19149, 49242, 105747, 200904, 349293, 567834, 875787, 1294752,...
91, 1932, 11193, 37996, 96915, 206976, 391657, 678888, 1101051, 1694980,...
136, 3512, 20176, 67936, 172328, 366616, 691792, 1196576, 1937416, 2978488,...
288, 5994, 33750, 112716, 284580, 603558, 1136394, 1962360, 3173256, 4873410,...
390, 9455, 53040, 176325, 443750, 939015, 1765080, 3044165, 4917750, 7546575,...
715, 14432, 79761, 263692, 661595, 1397220, 2622697, 4518536, 7293627,...
756, 20712, 115008, 379476, 950340, 2004216, 3758112, 6469428, 10435956,...
1508, 29614, 161538, 530348, 1324960, 2790138, 5226494, 8990488, 14494428,...
1722, 40243, 220024, 721245, 1799434, 3785467, 7085568, 12181309, 19629610,...
2835, 54420, 293985, 960300, 2391675, 5025960, 9400545, 16152360, 26017875,...
3088, 70800, 383904, 1252960, 3117648, 6546768, 12238240, 21019104,...
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Crossrefs

Cf. A367119 (first row), A367122 (second row), A135565 (first column), A367183 (vertices), A366253 (regions).

Sequence in context: A062834 A047980 A084702 * A168061 A261381 A365971

Adjacent sequences: A367187 A367188 A367189 * A367191 A367192 A367193

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Nov 09 2023

Status

approved