The $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets

TL;DR

This paper derives explicit formulas for the $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets, extending the understanding of generalized Frobenius problems for these special sequences.

Contribution

It provides the first explicit formulas for the $p$-Frobenius and $p$-Sylvester numbers for Fibonacci and Lucas triplets, advancing the study of generalized Frobenius problems.

Findings

Explicit formulas for Fibonacci $p$-Frobenius numbers

Explicit formulas for Lucas $p$-Frobenius numbers

Formulas for the total count of integers representable at most $p$ times

Abstract

In this paper we study a certain kind of generalized linear Diophantine problem of Frobenius. Let $a_1,a_2,\dots,a_l$ be positive integers such that their greatest common divisor is one. For a nonnegative integer $p$, denote the $p$-Frobenius number by $g_p(a_1,a_2,\dots,a_l)$, which is the largest integer that can be represented at most $p$ ways by a linear combination with nonnegative integer coefficients of $a_1,a_2,\dots,a_l$. When $p=0$, $0$-Frobenius number is the classical Frobenius number. When $l=2$, $p$-Frobenius number is explicitly given. However, when $l=3$ and even larger, even in special cases, it is not easy to give the Frobenius number explicitly, and it is even more difficult when $p>0$, and no specific example has been known. However, very recently, we have succeeded in giving explicit formulas for the case where the sequence is of triangular numbers or of repunits for the case where $l=3$. In this paper, we show the explicit formula for the Fibonacci triple when $p>0$. In addition, we give an explicit formula for the $p$-Sylvester number, that is, the total number of nonnegative integers that can be represented in at most $p$ ways. Furthermore, explicit formulas are shown concerning the Lucas triple.