The Frobenius Number for Jacobsthal Triples Associated with Number of Solutions

Abstract

In this paper, we find a formula for the largest integer (p-Frobenius number) such that a linear equation of non-negative integer coefficients composed of a Jacobsthal triplet has at most p representations. For p=0, the problem is reduced to the famous linear Diophantine problem of Frobenius, the largest integer of which is called the Frobenius number. We also give a closed formula for the number of non-negative integers (p-genus), such that linear equations have at most p representations. Extensions to the Jacobsthal polynomial and the Jacobsthal–Lucas polynomial give more general formulas that include the familiar Fibonacci and Lucas numbers. A basic problem with the Fibonacci triplet was dealt by Marin, Ramírez Alfonsín and M. P. Revuelta for p=0 and by Komatsu and Ying for the general non-negative integer p.

p-Numerical Semigroups of Generalized Fibonacci Triples

“…That is, they form a complete residue system modulo a, and each element is represented by a, va + b, vaJ k−1 (v) + bJ k (v) in at least p + 1 ways. The rough structure is similar to that in [18,19], though the structures of the p-Apéry set in other cases are not necessarily similar or have not been known yet.…”

“…So, the results in [18] are recovered as special cases. In addition, if v = 1, the results in [24,19] are recovered as special cases.…”

“…In this triple {a, va+b, vaJ k−1 (v)+bJ k (v)}, we can use the similar framework to that in [18] to construct the elements of the p-Apéry set. It is very important to see that such a framework is not always possible.…”

“…See [18,19] etc. for a detailed explanation that the elements located within the entire specified areas actually constitute the elements of the p-Apéry set.…”

“…Indeed, it is even more difficult when p > 0. However, finally, we have succeeded in obtaining p-Frobenius number in triangular numbers [15] and repunits [16] as well as Fibonacci and Lucas triplets [19] and Jacobsthal triples [18] quite recently.…”