This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A367190 #17 Nov 10 2023 02:04:28 %S A367190 3,24,8,153,124,20,588,780,390,42,1635,2816,2370,939,91,3708,7480, %T A367190 8300,5568,1932,136,7329,16428,21600,19149,11193,3512,288,13128,31724, %U A367190 46770,49242,37996,20176,5994,390,21843,55840,89390,105747,96915,67936,33750,9455,715 %N 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. %C A367190 "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 A367190 See A367183 and A366253 for images of the n-gons. %F A367190 T(n,k) = A367183(n,k) + A366253(n,k) - 1 by Euler's formula. %F A367190 Conjectures: %F A367190 T(3,k) = A367119(k) = (9/2)*k^4 + 6*k^3 + (9/2)*k^2 + 6*k + 3. %F A367190 T(4,k) = A367122(k) = 17*k^4 + 38*k^3 + 37*k^2 + 24*k + 8. %F A367190 T(5,k) = 45*k^4 + 120*k^3 + 130*k^2 + 75*k + 20. %F A367190 T(6,k) = (195/2)*k^4 + 285*k^3 + (657/2)*k^2 + 186*k + 42. %F A367190 T(7,k) = (371/2)*k^4 + 574*k^3 + (1379/2)*k^2 + 392*k + 91. %F A367190 T(8,k) = 322*k^4 + 1036*k^3 + 1282*k^2 + 736*k + 136. %F A367190 T(9,k) = 522*k^4 + 1728*k^3 + 2187*k^2 + 1269*k + 288. %F A367190 T(10,k) = (1605/2)*k^4 + 2715*k^3 + (6995/2)*k^2 + 2050*k + 390. %e A367190 The table begins: %e A367190 3, 24, 153, 588, 1635, 3708, 7329, 13128, 21843, 34320, 51513, 74484, 104403,... %e A367190 8, 124, 780, 2816, 7480, 16428, 31724, 55840, 91656, 142460, 211948, 304224,... %e A367190 20, 390, 2370, 8300, 21600, 46770, 89390, 156120, 254700, 393950, 583770,... %e A367190 42, 939, 5568, 19149, 49242, 105747, 200904, 349293, 567834, 875787, 1294752,... %e A367190 91, 1932, 11193, 37996, 96915, 206976, 391657, 678888, 1101051, 1694980,... %e A367190 136, 3512, 20176, 67936, 172328, 366616, 691792, 1196576, 1937416, 2978488,... %e A367190 288, 5994, 33750, 112716, 284580, 603558, 1136394, 1962360, 3173256, 4873410,... %e A367190 390, 9455, 53040, 176325, 443750, 939015, 1765080, 3044165, 4917750, 7546575,... %e A367190 715, 14432, 79761, 263692, 661595, 1397220, 2622697, 4518536, 7293627,... %e A367190 756, 20712, 115008, 379476, 950340, 2004216, 3758112, 6469428, 10435956,... %e A367190 1508, 29614, 161538, 530348, 1324960, 2790138, 5226494, 8990488, 14494428,... %e A367190 1722, 40243, 220024, 721245, 1799434, 3785467, 7085568, 12181309, 19629610,... %e A367190 2835, 54420, 293985, 960300, 2391675, 5025960, 9400545, 16152360, 26017875,... %e A367190 3088, 70800, 383904, 1252960, 3117648, 6546768, 12238240, 21019104,... %e A367190 . %e A367190 . %e A367190 . %Y A367190 Cf. A367119 (first row), A367122 (second row), A135565 (first column), A367183 (vertices), A366253 (regions). %K A367190 nonn,tabl %O A367190 3,1 %A A367190 _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 09 2023