A392282 - OEIS

%I #12 Jan 09 2026 23:25:22

%S 3,9,4,51,16,5,255,176,30,6,855,988,475,57,7,2193,3364,2720,1068,105,

%T 8,4719,8624,9225,6099,2107,184,9,8991,18496,23530,20550,11935,3776,

%U 306,10,15675,35116,50255,52161,39963,21200,6291,485,11,25545,61028,95100,111012,101017,70600,35028,9900,737,12

%N 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.

%C 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.

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

%C See A392228 and A392261 for images of the planar graphs.

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

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

%F 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.

%e The table begins:

%e 3, 9, 51, 255, 855, 2193, 4719, 8991, 15675, 25545, 39483, 58479, 83631, ...

%e 4, 16, 176, 988, 3364, 8624, 18496, 35116, 61028, 99184, 152944, 226076, 322756, ...

%e 5, 30, 475, 2720, 9225, 23530, 50255, 95100, 164845, 267350, 411555, 607480, ...

%e 6, 57, 1068, 6099, 20550, 52161, 111012, 209523, 362454, 586905, 902316, 1330467, ...

%e 7, 105, 2107, 11935, 39963, 101017, 214375, 403767, 697375, 1127833, 1732227, ...

%e 8, 184, 3776, 21200, 70600, 177848, 376544, 708016, 1221320, 1973240, 3028288, ...

%e 9, 306, 6291, 35028, 116109, 291654, 616311, 1157256, 1994193, 3219354, 4937499, ...

%e 10, 485, 9900, 54715, 180650, 452685, 955060, 1791275, 3084090, 4975525, 7626860, ...

%e 11, 737, 14883, 81719, 268895, 672441, 1416767, 2654663, 4567299, 7364225, ...

%e 12, 1080, 21552, 117660, 386028, 963672, 2028000, 3796812, 6528300, 10521048, ...

%e .

%e .

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

%K nonn,tabl

%O 3,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Jan 06 2026