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 A357235 #23 Sep 21 2022 12:00:32 %S A357235 3,6,4,15,8,5,30,20,10,6,51,40,25,12,7,66,68,50,30,14,8,111,88,85,60, %T A357235 35,16,9,150,148,130,102,70,40,18,10,171,168,185,156,119,80,45,20,11, %U A357235 246,260,250,222,182,136,90,50,22,12,303,296,325,300,259,208,153,100,55,24,13 %N A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices 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 A357235 Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon). %H A357235 Scott R. Shannon, <a href="/A357235/a357235.txt">Extended table for n = 3..50, k = 0..75</a>. %H A357235 Scott R. Shannon, <a href="/A357235/a357235.png">Image of T(5,20) = 2005</a>. %H A357235 Scott R. Shannon, <a href="/A357235/a357235_1.png">Image of T(7,10) = 707</a>. %F A357235 T(n,k) = A357254(n,k) - A357216(n,k) + 1 by Euler's formula. %F A357235 T(n,0) = n. %F A357235 T(n,1) = 2n. %F A357235 Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + n. %F A357235 T(3,k) = A357007(k). %F A357235 T(4,k) = A357060(k). %F A357235 T(6,k) = A357197(k). %F A357235 Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + n. %e A357235 The table begins: %e A357235 3, 6, 15, 30, 51, 66, 111, 150, 171, 246, 303, 312, 435, 510, 543, ... %e A357235 4, 8, 20, 40, 68, 88, 148, 168, 260, 296, 404, 436, 580, 632, 788, ... %e A357235 5, 10, 25, 50, 85, 130, 185, 250, 325, 410, 505, 610, 725, 850, 985, ... %e A357235 6, 12, 30, 60, 102, 156, 222, 300, 390, 468, 606, 708, 870, 1020, 1152, ... %e A357235 7, 14, 35, 70, 119, 182, 259, 350, 455, 574, 707, 854, 1015, 1190, 1379, ... %e A357235 8, 16, 40, 80, 136, 208, 296, 400, 520, 656, 808, 976, 1160, 1360, 1576, ... %e A357235 9, 18, 45, 90, 153, 234, 333, 450, 585, 738, 909, 1098, 1305, 1530, 1773, ... %e A357235 10, 20, 50, 100, 170, 260, 370, 500, 650, 820, 1010, 1220, 1450, 1700, 1970, ... %e A357235 11, 22, 55, 110, 187, 286, 407, 550, 715, 902, 1111, 1342, 1595, 1870, 2167, ... %e A357235 12, 24, 60, 120, 204, 312, 444, 600, 780, 984, 1212, 1464, 1740, 2040, 2364, ... %e A357235 13, 26, 65, 130, 221, 338, 481, 650, 845, 1066, 1313, 1586, 1885, 2210, 2561, ... %e A357235 14, 28, 70, 140, 238, 364, 518, 700, 910, 1148, 1414, 1708, 2030, 2380, 2758, ... %e A357235 15, 30, 75, 150, 255, 390, 555, 750, 975, 1230, 1515, 1830, 2175, 2550, 2955, ... %e A357235 See the attached text file for further examples. %e A357235 See A357007, A357060, A357197 for more images of the n-gons. %Y A357235 Cf. A357216 (regions), A357254 (edges), A357007 (triangle), A357060 (square), A357197 (hexagon), A007569, A146212. %K A357235 nonn,tabl %O A357235 3,1 %A A357235 _Scott R. Shannon_, Sep 19 2022