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A375986

Maximum number of edges in a simple polygon that is the regular union of n triangles.

3, 12, 22, 33, 45, 56, 67, 80, 91

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OFFSET

1,1

COMMENTS

For n >= 10, 11*n - 8 <= a(n) <= 12*n - 18 + [10 <= n <= 17] where [] is the Iverson bracket. - Giedrius Alkauskas, Dec 12 2025

A union of triangles is said to be regular if their vertices along the common border are arranged cyclically. If the union is non-regular, the maximal number of edges i(n) satisfies i(n) > c*A002004(n). So, it grows superlinearly. It is an open question to find n for which i(n) > a(n). - Giedrius Alkauskas, Jan 09 2026

LINKS

Table of n, a(n) for n=1..9.

Giedrius Alkauskas, Construction which shows that for n >= 4, a(n) >= 11*n-11.

Giedrius Alkauskas, Perfect unions of triangles.

Giedrius Alkauskas, An example which demonstrates that e(7) >= 67.

Giedrius Alkauskas, Triangle unions with maximal number of sides, arXiv:2510.22584 [math.CO], 2025.

Math Stackexchange, If a union of n triangles is a simple M-gon, find the maximum of M.

EXAMPLE

For n = 1, one has three vertices of a single triangle.

For n = 2, the example is a hexagram (the star of David).

CROSSREFS

Cf. A250001, A090338, A090339, A006066.