A367323 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.

1, 12, 4, 75, 56, 11, 252, 340, 170, 24, 715, 1120, 1161, 408, 50, 1572, 3264, 3900, 2268, 868, 80, 3109, 6264, 10741, 8208, 5594, 1488, 154, 5676, 13968, 22380, 20832, 18396, 9312, 2754, 220, 9291, 22904, 44491, 44640, 48462, 31552, 16858, 4220, 375

Offset

3,2

Comments

See A367322 and the cross references for further images of the n-gons.

Links

Table of n, a(n) for n=3..47.

Scott R. Shannon, Image for T(12,2).

Scott R. Shannon, Image for T(23,1).

Formula

T(n,k) = A367324(n,k) - A367322(n,k) + 1 (Euler).

Example

The table begins:
1, 12, 75, 252, 715, 1572, 3109, 5676, 9291, 14556, 22081, 32502, 44935, 62868, ...
4, 56, 340, 1120, 3264, 6264, 13968, 22904, 38748, 58256, 95656, 120960, ...
11, 170, 1161, 3900, 10741, 22380, 44491, 76610, 126336, 194070, 290651, ...
24, 408, 2268, 8208, 20832, 44640, 89214, 154752, 249906, 390012, 590658, ...
50, 868, 5594, 18396, 48462, 101794, 195714, 336504, 549704, 841890, 1249676, ...
80, 1488, 9312, 31552, 83432, 174816, 339816, 584176, 953416, 1463936, 2173976, ...
154, 2754, 16858, 55098, 142318, 298350, 568162, 975294, 1585666, 2426292, ...
220, 4220, 25220, 84280, 217800, 456640, 873090, 1501520, 2436020, 3736540, ...
375, 6732, 39887, 129492, 330903, 692648, 1311443, 2248840, 3645885, 5574756, ...
444, 9000, 52056, 178200, 462504, 963576, 1854432, 3180816, 5157612, 7906080, ...
781, 13962, 80783, 261222, 662663, 1385332, 2613521, 4478188, 7246331, ...
952, 18676, 107142, 352828, 891870, 1870876, 3525494, 6053768, 9778370, ...
1456, 25860, 146956, 474000, 1196116, 2498010, 4700776, 8050080, 13008106, ...
1696, 33152, 188000, 615328, 1547792, 3244000, 6095600, 10458560, 16876160, ...
2500, 44098, 247334, 795634, 1999762, 4173296, 7838004, 13416740, ...
2466, 52236, 302148, 991800, 2502000, 5229396, 9846234, ...
4029, 70604, 391781, 1258028, 3152101, 6574000, 12328417, ...
4500, 86240, 475800, 1546280, 3865240, 8085040, 15143880, ...
6175, 107562, 591655, 1897182, 4741633, 9883986, ...
6820, 129448, 706288, 2288880, 5703698, 11924132, ...
9086, 157412, 859718, 2753192, 6866858, 14307932, ...
9024, 181152, 1001640, 3254160, 8117304, ...
12926, 222850, 1209776, 3870250, 9636276, ...
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Crossrefs

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).

Sequence in context: A166206 A040137 A092237 * A081987 A327972 A047709

Adjacent sequences: A367320 A367321 A367322 * A367324 A367325 A367326

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Nov 14 2023

Status

approved