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 A357216 #23 Sep 21 2022 12:00:26 %S A357216 1,4,1,13,5,1,28,17,6,1,49,37,21,7,1,70,65,46,25,8,1,109,93,81,55,29, %T A357216 9,1,148,145,126,97,64,33,10,1,181,181,181,151,113,73,37,11,1,244,257, %U A357216 246,217,176,129,82,41,12,1,301,309,321,295,253,201,145,91,45,13,1 %N A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts. %C A357216 Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon). %H A357216 Scott R. Shannon, <a href="/A357216/a357216.txt">Extended table for n = 3..50, k = 0..75</a>. %H A357216 Scott R. Shannon, <a href="/A357216/a357216.jpg">Image of T(5,20) = 2001</a>. %H A357216 Scott R. Shannon, <a href="/A357216/a357216_1.jpg">Image of T(7,10) = 701</a>. %F A357216 T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula. %F A357216 T(n,0) = 1. %F A357216 T(n,1) = n + 1. %F A357216 Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1. %F A357216 T(3,k) = A356984(k). %F A357216 T(4,k) = A357058(k). %F A357216 T(6,k) = A357196(k). %F A357216 Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1. %e A357216 The table begins: %e A357216 1, 4, 13, 28, 49, 70, 109, 148, 181, 244, 301, 334, 433, 508, 565, ... %e A357216 1, 5, 17, 37, 65, 93, 145, 181, 257, 309, 401, 457, 577, 653, 785, ... %e A357216 1, 6, 21, 46, 81, 126, 181, 246, 321, 406, 501, 606, 721, 846, 981, ... %e A357216 1, 7, 25, 55, 97, 151, 217, 295, 385, 475, 601, 715, 865, 1015, 1159, ... %e A357216 1, 8, 29, 64, 113, 176, 253, 344, 449, 568, 701, 848, 1009, 1184, 1373, ... %e A357216 1, 9, 33, 73, 129, 201, 289, 393, 513, 649, 801, 969, 1153, 1353, 1569, ... %e A357216 1, 10, 37, 82, 145, 226, 325, 442, 577, 730, 901, 1090, 1297, 1522, 1765, ... %e A357216 1, 11, 41, 91, 161, 251, 361, 491, 641, 811, 1001, 1211, 1441, 1691, 1961, ... %e A357216 1, 12, 45, 100, 177, 276, 397, 540, 705, 892, 1101, 1332, 1585, 1860, 2157, ... %e A357216 1, 13, 49, 109, 193, 301, 433, 589, 769, 973, 1201, 1453, 1729, 2029, 2353, ... %e A357216 1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ... %e A357216 1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ... %e A357216 1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ... %e A357216 ... %e A357216 See the attached text file for further examples. %e A357216 See A356984, A357058, A357196 for more images of the n-gons. %Y A357216 Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857. %K A357216 nonn,tabl %O A357216 3,2 %A A357216 _Scott R. Shannon_, Sep 18 2022