A367304 Table read by antidiagonals: Place k equally spaced points on each side of a regular n-gon and join every pair of these n*k points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of regions in the resulting planar graph.

1, 4, 1, 27, 8, 1, 130, 88, 16, 1, 385, 444, 246, 30, 1, 1044, 1544, 1376, 492, 57, 1, 2005, 3584, 4621, 2814, 1079, 88, 1, 4060, 8356, 11691, 9042, 6014, 1800, 163, 1, 6831, 14996, 25026, 23604, 20049, 10016, 3196, 230, 1, 11272, 26572, 47386, 50448, 50597, 34432, 17632, 4770, 386, 1

Offset

3,2

Comments

See A367278 and A006533 for other images of the n-gons.

Links

Table of n, a(n) for n=3..57.

Scott R. Shannon, Image for T(5,5).

Scott R. Shannon, Image for T(7,4).

Scott R. Shannon, Image for T(8,4).

Scott R. Shannon, Image for T(12,3).

Formula

a(n,k) = A367305(n,k) - A367302(n,k) + 1 (Euler).

Example

The table begins:
1, 4, 27, 130, 385, 1044, 2005, 4060, 6831, 11272, 16819, 26436, 35737, 52147, ...
1, 8, 88, 444, 1544, 3584, 8356, 14996, 26572, 42144, 69988, 93264, 148364, ...
1, 16, 246, 1376, 4621, 11691, 25026, 47386, 82096, 133076, 204716, 301861, ...
1, 30, 492, 2814, 9042, 23604, 50448, 95244, 163890, 268848, 415146, 610476, ...
1, 57, 1079, 6014, 20049, 50597, 107171, 201916, 348559, 563375, 864977, ...
1, 88, 1800, 10016, 34432, 86360, 185856, 347976, 604248, 974184, 1502416, ...
1, 163, 3196, 17632, 58195, 146071, 308296, 578926, 997219, 1609453, 2467720, ...
1, 230, 4770, 26470, 89160, 222730, 474120, 887230, 1532880, 2470640, 3798120, ...
1, 386, 7525, 41053, 134729, 336678, 708753, 1327987, 2284151, 3682306, ...
1, 456, 9276, 56100, 187872, 468660, 1002300, 1873824, 3235104, 5214684, ...
1, 794, 15250, 82447, 269309, 670892, 1409630, 2637051, 4530891, ...
1, 966, 20286, 109956, 363552, 902174, 1904504, 3555020, 6119918, ...
1, 1471, 27811, 149266, 485761, 1207201, 2532751, 4732516, 8124511, ...
.
.
.

Crossrefs

Cf. A367302 (vertices), A367303 (internal vertices), A367305 (edges), A366486 (first row), A367278 (second row), A006533 (second column).

Sequence in context: A329060 A118283 A095891 * A095887 A225213 A137906

Adjacent sequences: A367301 A367302 A367303 * A367305 A367306 A367307

Keyword

nonn,tabl

Author

Scott R. Shannon, Nov 13 2023

Status

approved