%I #34 Nov 13 2023 07:29:19

%S 3,12,72,282,795,1818,3612,6492,10827,17040,25608,37062,51987,71022,

%T 94860,124248,159987,202932,253992,314130,384363,465762,559452,666612,

%U 788475,926328,1081512,1255422,1449507,1665270,1904268,2168112,2458467,2777052,3125640,3506058,3920187,4369962

%N 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.

%C "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.

%C 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.

%H Paolo Xausa, <a href="/A367117/b367117.txt">Table of n, a(n) for n = 0..10000</a>

%H Scott R. Shannon, <a href="/A367117/a367117.png">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A367117/a367117_1.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A367117/a367117_2.png">Image for n = 5</a>.

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

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

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

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

%Y See also A091908, A092098, A331782, A365929.

%Y If the boundary points are equally spaced, we get A274585, A092866, A274586, A092867. - _N. J. A. Sloane_, Nov 09 2023

%K nonn

%O 0,1

%A _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 05 2023.