A392261 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 vertices in the resulting planar graph.

3, 6, 4, 24, 9, 5, 120, 82, 15, 6, 411, 475, 225, 27, 7, 1068, 1644, 1325, 513, 49, 8, 2316, 4249, 4545, 2994, 1022, 86, 9, 4434, 9154, 11655, 10170, 5887, 1844, 144, 10, 7755, 17427, 24965, 25911, 19831, 10490, 3087, 230, 11, 12666, 30340, 47325, 55257, 50267, 35096, 17370, 4875, 352, 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.

Links

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

Scott R. Shannon, Illustration for T(3,5) = 1068.

Scott R. Shannon, Illustration for T(4,6) = 9154.

Scott R. Shannon, Illustration for T(5,3) = 1325.

Scott R. Shannon, Illustration for T(12,2) = 10662.

Scott R. Shannon, Proof of the general formulas given in A392261, A392282, A392228.

Formula

T(n,k) = A392282(n,k) - A392228(n,k) + 1 by Euler's formula.

T(3,k) = 3*A366478(k) = (9*k^4 - 12*k^3 + 3*k^2)/4 + 3*(k + 1).

T(n,k) = k^2*n*(n-1)*(k^2*(n^2+n-3) - 6*k*(n-1) + 3)/24 + n*(k + 1). See the attached file for proof.

Example

The array begins:
3, 6, 24, 120, 411, 1068, 2316, 4434, 7755, 12666, 19608, 29076, 41619, 57840, ...
4, 9, 82, 475, 1644, 4249, 9154, 17427, 30340, 49369, 76194, 112699, 160972, ...
5, 15, 225, 1325, 4545, 11655, 24965, 47325, 82125, 133295, 205305, 303165, ...
6, 27, 513, 2994, 10170, 25911, 55257, 104418, 180774, 292875, 450441, 664362, ...
7, 49, 1022, 5887, 19831, 50267, 106834, 201397, 348047, 563101, 865102,
8, 86, 1844, 10490, 35096, 88598, 187796, 353354, 609800, 985526, 1512788
9, 144, 3087, 17370, 57789, 145404, 307539, 577782, 995985, 1608264, 2466999
10, 230, 4875, 27175, 89990, 225810, 476755, 894575, 1540650, 2485990, 3811235, ...
11, 352, 7348, 40634, 134035, 335566, 707432, 1326028, 2281939, 3679940, ...
12, 519, 10662, 58557, 192516, 481047, 1012854, 1896837, 3262092, 5257911, ...
.
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Crossrefs

Cf. A366478 (n=3), A392173 (n=4), A392228 (regions), A392282 (edges), A367322.

Sequence in context: A145691 A245767 A367302 * A009782 A294670 A016615

Adjacent sequences: A392258 A392259 A392260 * A392262 A392263 A392264

Keyword

nonn,tabl

Author

Scott R. Shannon and N. J. A. Sloane, Jan 05 2026

Status

approved