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 A366253 #71 Nov 13 2023 07:29:26 %S A366253 1,13,4,82,67,11,307,406,206,24,841,1441,1216,489,50,1891,3796,4211, %T A366253 2835,995,80,3718,8299,10901,9672,5671,1802,154,6637,15982,23536, %U A366253 24780,19139,10196,3052,220,11017,28081,44906,53109,48686,34166,17011,4810,375 %N A366253 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 regions in the resulting planar graph. %C A366253 "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 A366253 Note that although the number of regions with a given number of edges in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices created from the edge-point chords remain simple. %H A366253 Scott R. Shannon, <a href="/A366253/a366253.png">Image for T(5,3)</a>. %H A366253 Scott R. Shannon, <a href="/A366253/a366253_1.png">Image for T(6,2)</a>. %H A366253 Scott R. Shannon, <a href="/A366253/a366253_2.png">Image for T(8,2)</a>. %H A366253 Scott R. Shannon, <a href="/A366253/a366253_3.png">Image for T(10,2)</a>. %F A366253 T(n,k) = A367190(n,k) - A367183(n,k) + 1 by Euler's formula. %F A366253 Conjectured: %F A366253 T(3,k) = A367118(k) = (9/4)*k^4 + 3*k^3 + (15/4)*k^2 + 3*k + 1. %F A366253 T(4,k) = A367121(k) = (17/2)*k^4 + 19*k^3 + (43/2)*k^2 + 14*k + 4. %F A366253 T(5,k) = (45/2)*k^4 + 60*k^3 + 70*k^2 + (85/2)*k + 11. %F A366253 T(6,k) = (195/4)*k^4 + (285/2)*k^3 + (687/4)*k^2 + 102*k + 24. %F A366253 T(7,k) = (371/4)*k^4 + 287*k^3 + (1421/4)*k^2 + 210*k + 50. %F A366253 T(8,k) = 161*k^4 + 518*k^3 + 655*k^2 + 388*k + 80. %F A366253 T(9,k) = 261*k^4 + 864*k^3 + (2223/2)*k^2 + (1323/2)*k + 154. %F A366253 T(10,k) = (1605/4)*k^4 + (2715/2)*k^3 + (7085/4)*k^2 + 1060*k + 220. %e A366253 The table begins: %e A366253 1, 13, 82, 307, 841, 1891, 3718, 6637, 11017, 17281, 25906, 37423, 52417,... %e A366253 4, 67, 406, 1441, 3796, 8299, 15982, 28081, 46036, 71491, 106294, 152497,... %e A366253 11, 206, 1216, 4211, 10901, 23536, 44906, 78341, 127711, 197426, 292436,... %e A366253 24, 489, 2835, 9672, 24780, 53109, 100779, 175080, 284472, 438585, 648219,... %e A366253 50, 995, 5671, 19139, 48686, 103825, 196295, 340061, 551314, 848471, 1252175,... %e A366253 80, 1802, 10196, 34166, 86480, 183770, 346532, 599126, 969776, 1490570,... %e A366253 154, 3052, 17011, 56611, 142696, 302374, 569017, 982261, 1588006, 2438416,... %e A366253 220, 4810, 26705, 88495, 222400, 470270, 883585, 1523455, 2460620, 3775450,... %e A366253 375, 7305, 40096, 132243, 331431, 699535, 1312620, 2260941, 3648943, 5595261,... %e A366253 444, 10509, 57810, 190263, 475980, 1003269, 1880634, 3236775, 5220588, 8001165,... %e A366253 781, 14938, 81082, 265747, 663391, 1396396, 2615068, 4497637, 7250257,... %e A366253 952, 20335, 110439, 361354, 900844, 1894347, 3544975, 6093514, 9818424,... %e A366253 1456, 27391, 147421, 480931, 1197076, 2514781, 4702741, 8079421, 13013056,... %e A366253 1696, 35716, 192552, 627484, 1560352, 3275556, 6122056, 10513372,... %e A366253 . %e A366253 . %e A366253 . %Y A366253 Cf. A367118 (first row), A367121 (second row), A007678 (first column), A367183 (vertices), A367190 (edges). %K A366253 nonn,tabl %O A366253 3,2 %A A366253 _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 09 2023