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A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.

%I A367276 #16 Nov 13 2023 18:41:51
%S A367276 4,9,69,345,1337,2885,7445,12833,23365,36589,64669,80133,138313,
%T A367276 176885,233765,312013,455273,513277,741965,819589,1046245,1310761,
%U A367276 1692961,1772097,2315289,2713997,3165125,3552753,4538845,4602985,6015561,6432681,7421345,8550485,9439621,10063993,12635769
%N A367276 Place n equally spaced points on each side of a square, and join each of these points by a chord to the 3*n new points on the other three sides: sequence gives number of vertices in the resulting planar graph.
%C A367276 We start with the four corner points of the square, 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 A367276 Each of the n points added to an edge is joined by 3*n chords to the points that were added to the other three edges. There are 6*n^2 chords.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276.png">Image for n = 1</a>.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276_1.png">Image for n = 2</a>.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276_2.png">Image for n = 3</a>.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276_3.png">Image for n = 4</a>.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276_4.png">Image for n = 5</a>.
%H A367276 Scott R. Shannon, <a href="/A367276/a367276_5.png">Image for n = 10</a>.
%F A367276 a(n) = A367279(n) - A367278(n) + 1 (Euler).
%Y A367276 Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).
%Y A367276 If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.
%Y A367276 If the 4*n points are placed uniformly and we also draw chords from the four corner points of the square to these 4*n points, we get A255011, A331448, A331449, A334690.
%K A367276 nonn
%O A367276 0,1
%A A367276 _Scott R. Shannon_, Nov 11 2023