login
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.
5

%I #16 Nov 13 2023 18:41:51

%S 4,9,69,345,1337,2885,7445,12833,23365,36589,64669,80133,138313,

%T 176885,233765,312013,455273,513277,741965,819589,1046245,1310761,

%U 1692961,1772097,2315289,2713997,3165125,3552753,4538845,4602985,6015561,6432681,7421345,8550485,9439621,10063993,12635769

%N 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 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 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 Scott R. Shannon, <a href="/A367276/a367276.png">Image for n = 1</a>.

%H Scott R. Shannon, <a href="/A367276/a367276_1.png">Image for n = 2</a>.

%H Scott R. Shannon, <a href="/A367276/a367276_2.png">Image for n = 3</a>.

%H Scott R. Shannon, <a href="/A367276/a367276_3.png">Image for n = 4</a>.

%H Scott R. Shannon, <a href="/A367276/a367276_4.png">Image for n = 5</a>.

%H Scott R. Shannon, <a href="/A367276/a367276_5.png">Image for n = 10</a>.

%F a(n) = A367279(n) - A367278(n) + 1 (Euler).

%Y Cf. A367277 (interior vertices), A367278 (regions), A367279 (edges).

%Y If the 4*n points are placed "in general position" instead of uniformly, we get sequences A334698, A367121, A367122.

%Y 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 nonn

%O 0,1

%A _Scott R. Shannon_, Nov 11 2023