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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.
4
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
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, ...
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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
KEYWORD
nonn,tabl
AUTHOR
Scott R. Shannon, Nov 13 2023
STATUS
approved