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A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides: sequence gives number of vertices in the resulting planar graph.

%I A366483 #45 Jul 09 2025 05:02:51
%S A366483 3,6,22,108,300,919,1626,3558,5824,9843,14352,23845,30951,47196,62773,
%T A366483 82488,104544,144784,173694,230008,276388,336927,403452,509218,582417,
%U A366483 702228,824956,969387,1098312,1321978,1463580,1724190,1952509,2221497,2505169,2846908,3103788,3556143,3978763,4444003
%N A366483 Place n equally spaced points on each side of an equilateral triangle, and join each of these points by a chord to the 2*n new points on the other two sides:  sequence gives number of vertices in the resulting planar graph.
%C A366483 We start with the three corner points of the triangle, and add n further points along each edge. Including the corner points, we end up with n+2 points along each edge, and the edge is divided into n+1 line segments.
%C A366483 Each of the n points added to an edge is joined by 2*n chords to the points that were added to the other two edges. There are 3*n^2 chords.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483.png">Image for n = 1</a>.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483_1.png">Image for n = 2</a>.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483_2.png">Image for n = 3</a>.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483_3.png">Image for n = 4</a>.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483_4.png">Image for n = 5</a>.
%H A366483 Scott R. Shannon, <a href="/A366483/a366483_5.png">Image for n = 10</a>.
%F A366483 a(n) = A366485(n) - A366486(n) + 1 (Euler).
%Y A366483 Cf. A366484 (interior vertices), A366485 (edges), A366486 (regions).
%Y A366483 If the 3*n points are placed "in general position" instead of uniformly, we get sequences A366478, A365929, A366932, A367015.
%Y A366483 If the 3*n points are placed uniformly and we also draw chords from the three corner points of the triangle to these 3*n points, we get A274585, A092866, A274586, A092867.
%K A366483 nonn
%O A366483 0,1
%A A366483 _Scott R. Shannon_ and _N. J. A. Sloane_, Nov 09 2023