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%I #22 Sep 21 2022 12:00:37
%S 3,9,4,27,12,5,57,36,15,6,99,76,45,18,7,135,132,95,54,21,8,219,180,
%T 165,114,63,24,9,297,292,255,198,133,72,27,10,351,348,365,306,231,152,
%U 81,30,11,489,516,495,438,357,264,171,90,33,12,603,604,645,594,511,408,297,190,99,36,13
%N Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of edges 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 Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).
%H Scott R. Shannon, <a href="/A357254/a357254.txt">Extended table for n = 3..50, k = 0..75</a>.
%F T(n,k) = A357216(n,k) + A357235(n,k) - 1 by Euler's formula.
%F T(n,0) = n.
%F T(n,1) = 3n.
%F Conjectured formula for all columns for n >= 7: T(n,k) = 2n*k^2 + n.
%F T(3,k) = A357008(k).
%F T(4,k) = A357061(k).
%F T(6,k) = A357198(k).
%F Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = 2n*k^2 + n.
%e The table begins:
%e 3, 9, 27, 57, 99, 135, 219, 297, 351, 489, 603, 645, 867, 1017, ...
%e 4, 12, 36, 76, 132, 180, 292, 348, 516, 604, 804, 892, 1156, 1284, ...
%e 5, 15, 45, 95, 165, 255, 365, 495, 645, 815, 1005, 1215, 1445, 1695, ...
%e 6, 18, 54, 114, 198, 306, 438, 594, 774, 942, 1206, 1422, 1734, 2034, ...
%e 7, 21, 63, 133, 231, 357, 511, 693, 903, 1141, 1407, 1701, 2023, 2373, ...
%e 8, 24, 72, 152, 264, 408, 584, 792, 1032, 1304, 1608, 1944, 2312, 2712, ...
%e 9, 27, 81, 171, 297, 459, 657, 891, 1161, 1467, 1809, 2187, 2601, 3051, ...
%e 10, 30, 90, 190, 330, 510, 730, 990, 1290, 1630, 2010, 2430, 2890, 3390, ...
%e 11, 33, 99, 209, 363, 561, 803, 1089, 1419, 1793, 2211, 2673, 3179, 3729, ...
%e 12, 36, 108, 228, 396, 612, 876, 1188, 1548, 1956, 2412, 2916, 3468, 4068, ...
%e 13, 39, 117, 247, 429, 663, 949, 1287, 1677, 2119, 2613, 3159, 3757, 4407, ...
%e 14, 42, 126, 266, 462, 714, 1022, 1386, 1806, 2282, 2814, 3402, 4046, 4746, ...
%e 15, 45, 135, 285, 495, 765, 1095, 1485, 1935, 2445, 3015, 3645, 4335, 5085, ...
%e ...
%e See the attached text file for further examples.
%e See A356984, A357058, A357196 for images of the n-gons.
%Y Cf. A357216 (regions), A357235 (vertices), A357008 (triangle), A357061 (square), A357198 (hexagon), A356984, A357058, A357196, A135565, A344899.
%K nonn,tabl
%O 3,1
%A _Scott R. Shannon_, Sep 20 2022
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