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

1, 70, 349, 916, 1474, 2296, 3412, 4978, 7042, 9748, 13132, 17506, 22786, 29410, 37288, 46630, 57574

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..16.

Scott R. Shannon, Image for the figure with edge-count coloring for n=1.

Scott R. Shannon, Image for the figure with edge-count coloring for n=2.

Scott R. Shannon, Image for the figure with edge-count coloring for n=3.

Scott R. Shannon, Image for the figure with edge-count coloring for n=4.

Scott R. Shannon, Image for the figure with edge-count coloring for n=5.

Scott R. Shannon, Image for the figure with edge-count coloring for n=6.

Example

a(0) = 1. There is one region in an equilateral triangle with no other polygons.
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.

Crossrefs

Cf. A337790 (number of vertices), A331456, A331452, A306302, A092867, A007678.

Sequence in context: A072596 A309310 A330702 * A174533 A174534 A295770

Adjacent sequences: A335858 A335859 A335860 * A335862 A335863 A335864

Keyword

nonn,more

Author

Scott R. Shannon and N. J. A. Sloane, Sep 22 2020

Status

approved