A392282 Array read by antidiagonals: Place k points in general position on each side of a regular n-gon and join every pair of the k*n boundary points by a chord; T(n,k) (n >= 3, k >= 0) gives the number of edges in the resulting planar graph.

3, 9, 4, 51, 16, 5, 255, 176, 30, 6, 855, 988, 475, 57, 7, 2193, 3364, 2720, 1068, 105, 8, 4719, 8624, 9225, 6099, 2107, 184, 9, 8991, 18496, 23530, 20550, 11935, 3776, 306, 10, 15675, 35116, 50255, 52161, 39963, 21200, 6291, 485, 11, 25545, 61028, 95100, 111012, 101017, 70600, 35028, 9900, 737, 12

Offset

3,1

Comments

There are k points in general position along the interior of each edge, the vertices of the n-gon themselves are not end-points of chords.

"In general position" implies that there is no point in the interior of the n-gon where three or more chords meet.

See A392228 and A392261 for images of the planar graphs.

Links

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

Formula

T(n,k) = A392228(n,k) + A392261(n,k) - 1 by Euler's formula.

T(3,k) = A366932(k) = (9/2)*k^4 - 6*k^3 + (9/2)*k^2 + 3*k + 3.

T(n,k) = k^2*n*(n-1)*(k^2*(n^2+n-3) - 6*k*(n-1) + 9)/12 + n*(k + 1). See A392261 for a proof.

Example

The table begins:
3, 9, 51, 255, 855, 2193, 4719, 8991, 15675, 25545, 39483, 58479, 83631, ...
4, 16, 176, 988, 3364, 8624, 18496, 35116, 61028, 99184, 152944, 226076, 322756, ...
5, 30, 475, 2720, 9225, 23530, 50255, 95100, 164845, 267350, 411555, 607480, ...
6, 57, 1068, 6099, 20550, 52161, 111012, 209523, 362454, 586905, 902316, 1330467, ...
7, 105, 2107, 11935, 39963, 101017, 214375, 403767, 697375, 1127833, 1732227, ...
8, 184, 3776, 21200, 70600, 177848, 376544, 708016, 1221320, 1973240, 3028288, ...
9, 306, 6291, 35028, 116109, 291654, 616311, 1157256, 1994193, 3219354, 4937499, ...
10, 485, 9900, 54715, 180650, 452685, 955060, 1791275, 3084090, 4975525, 7626860, ...
11, 737, 14883, 81719, 268895, 672441, 1416767, 2654663, 4567299, 7364225, ...
12, 1080, 21552, 117660, 386028, 963672, 2028000, 3796812, 6528300, 10521048, ...
.
.

Crossrefs

Cf. A366932 (n=3), A392174 (n=4), A392228 (regions), A392261 (vertices), A367324.

Sequence in context: A370464 A357254 A367305 * A258580 A021966 A386002

Adjacent sequences: A392279 A392280 A392281 * A392283 A392284 A392285

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Jan 06 2026

Status

approved