cp's OEIS Frontend

A335861 Number of regions in a Y-shaped polygon with equal arms of length n (see the Comments for definition).

%I A335861 #39 Sep 24 2020 22:09:29
%S A335861 1,70,349,916,1474,2296,3412,4978,7042,9748,13132,17506,22786,29410,
%T A335861 37288,46630,57574
%N A335861 Number of regions in a Y-shaped polygon with equal arms of length n (see the Comments for definition).
%C A335861 This polygon consists of a central equilateral triangle with a line of n adjacent squares connected to each of its three edges. This gives the polygon a total of one triangle, 3n squares, and 6n+3 vertices. Join every pair of vertices by a line segment, provided the line does not extend beyond the boundary of the polygon. The sequence gives the number of regions in the resulting figure.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861.png">Image for the figure with edge-count coloring for n=1</a>.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861_1.png">Image for the figure with edge-count coloring for n=2</a>.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861_2.png">Image for the figure with edge-count coloring for n=3</a>.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861_3.png">Image for the figure with edge-count coloring for n=4</a>.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861_4.png">Image for the figure with edge-count coloring for n=5</a>.
%H A335861 Scott R. Shannon, <a href="/A335861/a335861_5.png">Image for the figure with edge-count coloring for n=6</a>.
%e A335861 a(0) = 1. There is one region in an equilateral triangle with no other polygons.
%e A335861 a(1) = 70. With one square adjacent to each of the triangles sides the resulting line segments form 48 triangles, twelves 4-gons, nine 5-gons, and one 6-gon. This gives a total of 70 regions. See the first linked image.
%Y A335861 Cf. A337790 (number of vertices), A331456, A331452, A306302, A092867, A007678.
%K A335861 nonn,more
%O A335861 0,2
%A A335861 _Scott R. Shannon_ and _N. J. A. Sloane_, Sep 22 2020