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A328526 Number of regions in an equilateral triangle "frame" of size n. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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
Scott R. Shannon, Illustration for n=7.
CROSSREFS
Cf. A333030 (edges), A333031 (vertices), A333032 (3-gons), A333033 (4-gons), A331776 (square frame), A092867 (filled triangle).
Sequence in context: A064116 A003368 A246767 * A092867 A292532 A053310
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(12) and beyond from Lars Blomberg, May 01 2020
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)