A367303 - OEIS

A367303

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 internal vertices in the resulting planar graph.

0, 0, 0, 13, 1, 0, 96, 57, 5, 0, 285, 329, 200, 13, 0, 901, 1317, 1265, 379, 35, 0, 1605, 2861, 4410, 2437, 987, 49, 0, 3534, 7417, 11280, 7843, 5789, 1577, 126, 0, 5797, 12801, 24455, 21235, 19642, 9193, 3042, 161, 0, 9813, 23329, 46570, 46141, 49931, 33161, 17280, 4371, 330, 0

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

3,4

COMMENTS

See A366484 and A367277 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(8,3).

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

FORMULA

a(n,k) = A367305(n,k) - A367304(n,k) + 1 - n*(k+1) (Euler).

EXAMPLE

The table begins:

0, 0, 13, 96, 285, 901, 1605, 3534, 5797, 9813, 14319, 23809, 30912, 47154, ...

0, 1, 57, 329, 1317, 2861, 7417, 12801, 23329, 36549, 64625, 80085, 138261, ...

0, 5, 200, 1265, 4410, 11280, 24455, 46570, 80960, 131510, 202555, 298630, ...

0, 13, 379, 2437, 7843, 21235, 46141, 87427, 150391, 249925, 388819, 569767, ...

0, 35, 987, 5789, 19642, 49931, 106120, 200508, 346619, 560602, 861252, ...

0, 49, 1577, 9193, 33161, 83313, 182649, 341369, 597097, 961681, 1490601, ...

0, 126, 3042, 17280, 57510, 145008, 306621, 576711, 994149, 1605267, 2462013, ...

0, 161, 4371, 25161, 87251, 218151, 469331, 877211, 1522141, 2452131, 3781431, ...

0, 330, 7293, 40524, 133738, 335126, 706310, 1324763, 2279695, 3676321, ...

0, 301, 7861, 53077, 182653, 456181, 990145, 1849453, 3207217, 5171377, ...

0, 715, 14924, 81705, 267930, 668733, 1406275, 2632604, 4524793, ...

0, 757, 19447, 107101, 360319, 893033, 1896567, 3536275, 6103763, ...

0, 1365, 27375, 148275, 483930, 1204305, 2528340, 4726650, 8116515, ...

CROSSREFS

Cf. A367302 (vertices), A367304 (regions), A367305 (edges), A366484 (first row), A367277 (second row).

Sequence in context: A180265 A165400 A181154 * A357912 A251072 A353805

Adjacent sequences: A367300 A367301 A367302 * A367304 A367305 A367306

KEYWORD

nonn,tabl

AUTHOR

Scott R. Shannon, Nov 13 2023

STATUS

approved