The number of convex pentagons and hexagons in an $n$-triangular net

TL;DR

This paper derives formulas to count convex pentagons and hexagons within an n-triangular net by solving recursive relations, advancing combinatorial geometry understanding.

Contribution

It provides explicit counting formulas for convex pentagons and hexagons in n-triangular nets, a novel combinatorial geometry result.

Findings

Derived formulas for convex pentagons in n-triangular nets

Derived formulas for convex hexagons in n-triangular nets

Solved recursive relations to obtain counting formulas

Abstract

In this paper, we obtain the counting formulaes of convex pentagons and convex hexagons, respectively, in an $n$-triangular net by solving the corresponding recursive formulaes.