A328526 Number of regions in an equilateral triangle "frame" of size n.

1, 12, 75, 249, 543, 1023, 1746, 2814, 4293, 6267, 8868, 12228, 16464, 21774, 28176, 35832, 45066, 56040, 68931, 84033, 101307, 120987, 143574, 169290, 198222, 230790, 267117, 307455, 352437, 402255, 457182, 517986, 584454, 656874, 735708, 821076, 913860

Offset: 1

Author

Scott R. Shannon and N. J. A. Sloane, Feb 24 2020

Keywords

• nonn

Comments

A equilateral triangular "frame" of size n is formed from a triangular grid consisting of an outer edge of (n+1) points with the central grid of (n-5)*(n-6)/2 points removed. If n is less than 4 then no points or triangles are removed, and a(n) = A092867(n). From now on we assume n >= 4.

If we focus on the triangles rather than the points, the frame consists of a grid of equilateral triangles with the central block of (n-3)^2 triangles removed.

The resulting structure has an outer perimeter with 3*n points and an inner perimeter with 3*n-9 points, for a total of 6*n-9 perimeter points. The frame itself is the strip equilateral triangles pointing in alternate directions between the inner and outer perimeters such that the frame thickness equals the height of one triangle.

Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.

Like the square frame of A331776 only regions with 3 or 4 edges are formed.

Links

• Lars Blomberg, Table of n, a(n) for n = 1..65
• Scott R. Shannon, Illustration for n=7.
• Scott R. Shannon, Illustration for n=4 with random distance-based coloring.
• Scott R. Shannon, Illustration for n=7 with random distance-based coloring.
• Scott R. Shannon, Illustration for n=11 with random distance-based coloring.

Crossrefs

Cf. A333030 (edges), A333031 (vertices), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A092867 (filled triangle).

Extensions

a(12) and beyond from Lars Blomberg, May 01 2020