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 A367183 #20 Nov 13 2023 18:40:55 %S A367183 3,12,5,72,58,10,282,375,185,19,795,1376,1155,451,42,1818,3685,4090, %T A367183 2734,938,57,3612,8130,10700,9478,5523,1711,135,6492,15743,23235, %U A367183 24463,18858,9981,2943,171,10827,27760,44485,52639,48230,33771,16740,4646,341 %N 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. %C A367183 "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. %H A367183 Scott R. Shannon, <a href="/A367183/a367183.png">Image for T(5,3)</a>. %H A367183 Scott R. Shannon, <a href="/A367183/a367183_1.png">Image for T(6,2)</a>. %H A367183 Scott R. Shannon, <a href="/A367183/a367183_2.png">Image for T(7,1)</a>. %H A367183 Scott R. Shannon, <a href="/A367183/a367183_3.png">Image for T(8,1)</a>. %F A367183 T(n,k) = A367190(n,k) - A366253(n,k) + 1 by Euler's formula. %F A367183 T(3,k) = A367117(k) = (9/4)*k^4 + 3*k^3 + (3/4)*k^2 + 3*k + 3. %F A367183 Conjectured: %F A367183 T(4,k) = A334698(k+1) = (17/2)*k^4 + 19*k^3 + (31/2)*k^2 + 10*k + 5. %F A367183 T(5,k) = (45/2)*k^4 + 60*k^3 + 60*k^2 + (65/2)*k + 10. %F A367183 T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (627/4)*k^2 + 84*k + 19. %F A367183 T(7,k) = (371/4)*k^4 + 287*k^3 + (1337/4)*k^2 + 182*k + 42. %F A367183 T(8,k) = 161*k^4 + 518*k^3 + 627*k^2 + 348*k + 57. %F A367183 T(9,k) = 261*k^4 + 864*k^3 + (2151/2)*k^2 + (1215/2)*k + 135. %F A367183 T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (6905/4)*k^2 + 990*k + 171. %e A367183 The table begins: %e A367183 3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987,... %e A367183 5, 58, 375, 1376, 3685, 8130, 15743, 27760, 45621, 70970, 105655, 151728,... %e A367183 10, 185, 1155, 4090, 10700, 23235, 44485, 77780, 126990, 196525, 291335,... %e A367183 19, 451, 2734, 9478, 24463, 52639, 100126, 174214, 283363, 437203, 646534,... %e A367183 42, 938, 5523, 18858, 48230, 103152, 195363, 338828, 549738, 846510, 1249787,... %e A367183 57, 1711, 9981, 33771, 85849, 182847, 345261, 597451, 967641, 1487919, 2194237,... %e A367183 135, 2943, 16740, 56106, 141885, 301185, 567378, 980100, 1585251, 2434995,... %e A367183 171, 4646, 26336, 87831, 221351, 468746, 881496, 1520711, 2457131, 3771126,... %e A367183 341, 7128, 39666, 131450, 330165, 697686, 1310078, 2257596, 3644685, 5589980,... %e A367183 313, 10204, 57199, 189214, 474361, 1000948, 1877479, 3232654, 5215369, 7994716,... %e A367183 728, 14677, 80457, 264602, 661570, 1393743, 2611427, 4492852, 7244172,... %e A367183 771, 19909, 109586, 359892, 898591, 1891121, 3540594, 6087796, 9811187,... %e A367183 1380, 27030, 146565, 479370, 1194600, 2511180, 4697805, 8072940, 13004820,... %e A367183 1393, 35085, 191353, 625477, 1557297, 3271213, 6116185, 10505733,...... %e A367183 . %e A367183 . %e A367183 . %Y A367183 Cf. A367117 (first row), A334698 (second row), A007569 (first column), A366253 (regions), A367190 (edges). %K A367183 nonn,tabl %O A367183 3,1 %A A367183 _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 08 2023