A367117 Place n points in general position on each side of an equilateral triangle, and join every pair of the 3*n+3 boundary points by a chord; sequence gives number of vertices in the resulting planar graph.

3, 12, 72, 282, 795, 1818, 3612, 6492, 10827, 17040, 25608, 37062, 51987, 71022, 94860, 124248, 159987, 202932, 253992, 314130, 384363, 465762, 559452, 666612, 788475, 926328, 1081512, 1255422, 1449507, 1665270, 1904268, 2168112, 2458467, 2777052, 3125640, 3506058, 3920187, 4369962

Offset

0,1

Comments

"In general position" implies that the internal lines (or chords) only have simple intersections. There is no interior point where three or more chords meet.

Note that although the number of k-gons in the graph will vary as the edge points change position, the total number of regions will stay constant as long as all internal vertices remain simple.

Links

Paolo Xausa, Table of n, a(n) for n = 0..10000

Scott R. Shannon, Image for n = 1.

Scott R. Shannon, Image for n = 2.

Scott R. Shannon, Image for n = 5.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

Theorem: a(n) = (3/4)*(n+1)*(3*n^3+n^2+4).

a(n) = A367119(n) - A367118(n) + 1 by Euler's formula.

G.f.: 3*(1 - x + 14*x^2 + 4*x^3)/(1 - x)^5. - Andrew Howroyd, Nov 13 2025

Mathematica

A367117[n_]:=3/4(n+1)(3n^3+n^2+4); Array[A367117, 50, 0] (* Paolo Xausa, Nov 09 2023 *)

Crossrefs

Cf. A367118 (regions), A367119 (edges).

See also A091908, A092098, A331782, A365929.

If the boundary points are equally spaced, we get A274585, A092866, A274586, A092867. - N. J. A. Sloane, Nov 09 2023

Sequence in context: A103366 A335643 A277457 * A228386 A175836 A293138

Adjacent sequences: A367114 A367115 A367116 * A367118 A367119 A367120

Keyword

nonn,easy

Author

Scott R. Shannon and N. J. A. Sloane, Nov 05 2023

Status

approved