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%I #15 Jan 06 2024 12:01:25
%S 1,12,4,75,56,11,252,340,170,24,715,1120,1161,408,50,1572,3264,3900,
%T 2268,868,80,3109,6264,10741,8208,5594,1488,154,5676,13968,22380,
%U 20832,18396,9312,2754,220,9291,22904,44491,44640,48462,31552,16858,4220,375
%N Table read by antidiagonals: Place k equally spaced points 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 the number of regions in the resulting planar graph.
%C See A367322 and the cross references for further images of the n-gons.
%H Scott R. Shannon, <a href="/A367323/a367323.png">Image for T(12,2)</a>.
%H Scott R. Shannon, <a href="/A367323/a367323_1.png">Image for T(23,1)</a>.
%F T(n,k) = A367324(n,k) - A367322(n,k) + 1 (Euler).
%e The table begins:
%e 1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, ...
%e 4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, ...
%e 11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, ...
%e 24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, ...
%e 50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, ...
%e 80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, ...
%e 154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, ...
%e 220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, ...
%e 375, 6732, 39887, 129492, 330903, 692648, 1311443, 2248840, 3645885, 5574756, ...
%e 444, 9000, 52056, 178200, 462504, 963576, 1854432, 3180816, 5157612, 7906080, ...
%e 781, 13962, 80783, 261222, 662663, 1385332, 2613521, 4478188, 7246331, ...
%e 952, 18676, 107142, 352828, 891870, 1870876, 3525494, 6053768, 9778370, ...
%e 1456, 25860, 146956, 474000, 1196116, 2498010, 4700776, 8050080, 13008106, ...
%e 1696, 33152, 188000, 615328, 1547792, 3244000, 6095600, 10458560, 16876160, ...
%e 2500, 44098, 247334, 795634, 1999762, 4173296, 7838004, 13416740, ...
%e 2466, 52236, 302148, 991800, 2502000, 5229396, 9846234, ...
%e 4029, 70604, 391781, 1258028, 3152101, 6574000, 12328417, ...
%e 4500, 86240, 475800, 1546280, 3865240, 8085040, 15143880, ...
%e 6175, 107562, 591655, 1897182, 4741633, 9883986, ...
%e 6820, 129448, 706288, 2288880, 5703698, 11924132, ...
%e 9086, 157412, 859718, 2753192, 6866858, 14307932, ...
%e 9024, 181152, 1001640, 3254160, 8117304, ...
%e 12926, 222850, 1209776, 3870250, 9636276, ...
%e .
%e .
%e .
%Y Cf. A367322 (vertices), A367324 (edges), A092867 (1st row), A255011 (2nd row), A331929 (3rd row), A331931 (4th row), A329713 (5th row), A333075 (6th row), A332421 (7th row), A333139 (8th row), A007678 (1st column).
%K nonn,tabl
%O 3,2
%A _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 14 2023
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