%I #17 Nov 10 2023 02:04:28
%S 3,24,8,153,124,20,588,780,390,42,1635,2816,2370,939,91,3708,7480,
%T 8300,5568,1932,136,7329,16428,21600,19149,11193,3512,288,13128,31724,
%U 46770,49242,37996,20176,5994,390,21843,55840,89390,105747,96915,67936,33750,9455,715
%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 edges 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 See A367183 and A366253 for images of the n-gons.
%F T(n,k) = A367183(n,k) + A366253(n,k) - 1 by Euler's formula.
%F Conjectures:
%F T(3,k) = A367119(k) = (9/2)*k^4 + 6*k^3 + (9/2)*k^2 + 6*k + 3.
%F T(4,k) = A367122(k) = 17*k^4 + 38*k^3 + 37*k^2 + 24*k + 8.
%F T(5,k) = 45*k^4 + 120*k^3 + 130*k^2 + 75*k + 20.
%F T(6,k) = (195/2)*k^4 + 285*k^3 + (657/2)*k^2 + 186*k + 42.
%F T(7,k) = (371/2)*k^4 + 574*k^3 + (1379/2)*k^2 + 392*k + 91.
%F T(8,k) = 322*k^4 + 1036*k^3 + 1282*k^2 + 736*k + 136.
%F T(9,k) = 522*k^4 + 1728*k^3 + 2187*k^2 + 1269*k + 288.
%F T(10,k) = (1605/2)*k^4 + 2715*k^3 + (6995/2)*k^2 + 2050*k + 390.
%e The table begins:
%e 3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403,...
%e 8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224,...
%e 20, 390, 2370, 8300, 21600, 46770, 89390, 156120, 254700, 393950, 583770,...
%e 42, 939, 5568, 19149, 49242, 105747, 200904, 349293, 567834, 875787, 1294752,...
%e 91, 1932, 11193, 37996, 96915, 206976, 391657, 678888, 1101051, 1694980,...
%e 136, 3512, 20176, 67936, 172328, 366616, 691792, 1196576, 1937416, 2978488,...
%e 288, 5994, 33750, 112716, 284580, 603558, 1136394, 1962360, 3173256, 4873410,...
%e 390, 9455, 53040, 176325, 443750, 939015, 1765080, 3044165, 4917750, 7546575,...
%e 715, 14432, 79761, 263692, 661595, 1397220, 2622697, 4518536, 7293627,...
%e 756, 20712, 115008, 379476, 950340, 2004216, 3758112, 6469428, 10435956,...
%e 1508, 29614, 161538, 530348, 1324960, 2790138, 5226494, 8990488, 14494428,...
%e 1722, 40243, 220024, 721245, 1799434, 3785467, 7085568, 12181309, 19629610,...
%e 2835, 54420, 293985, 960300, 2391675, 5025960, 9400545, 16152360, 26017875,...
%e 3088, 70800, 383904, 1252960, 3117648, 6546768, 12238240, 21019104,...
%e .
%e .
%e .
%Y Cf. A367119 (first row), A367122 (second row), A135565 (first column), A367183 (vertices), A366253 (regions).
%K nonn,tabl
%O 3,1
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 09 2023