The Frobenius number for sequences of triangular numbers associated with number of solutions

TL;DR

This paper derives explicit formulas for the generalized Frobenius numbers of triples of triangular numbers, extending known results and addressing cases where the number of representations is at most a given integer.

Contribution

It provides the first explicit formulas for the generalized Frobenius numbers for triples of triangular numbers, generalizing previous results for the case p=0.

Findings

Explicit formulas for the Frobenius numbers of triples of triangular numbers.

Extension of known results to cases where the number of representations is at most p.

Addresses a longstanding open problem in the Frobenius number literature.

Abstract

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of $a_1,\dots,a_k$ is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer $p$. It is easy to find the explicit form of this number in the case of two variables. However, no explicit form has been known even in any special case of three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When $p=0$, their Frobenius number is given by Robles-P\'erez and Rosales in 2018.