The Frobenius number for the triple of the 2-step star numbers

TL;DR

This paper derives explicit formulas for the Frobenius and Sylvester numbers for triples of 2-step star numbers, extending understanding of their number-theoretic properties and variations of classical star numbers.

Contribution

It provides the first closed-form expressions for Frobenius and Sylvester numbers for these specific triples of 2-step star numbers.

Findings

Closed-form formulas for Frobenius numbers.

Closed-form formulas for Sylvester numbers.

Extension of classical star number properties.

Abstract

In this paper, we give closed form expressions of the Frobenius number for the triple of the $2$-step star numbers $an(n-2) + 1$ for an integer $a \geq 4$. These numbers have been studied from different aspects for some $a$'s. These numbers can also be considered as variations of the well known star numbers of the form $6n(n-1) + 1$. We also give closed form expressions of the Sylvester number (genus) for the triple of the $2$-step star numbers.