On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II

Abstract

For a set $A$ of positive integers with $\gcd (A)=1$, let $⟨A⟩$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. Then it is well known that only finitely many positive integers do not belong to $⟨A⟩$. The Frobenius number and the genus associated with the set $A$ is the largest number and the cardinality of the set of integers non-representable by $A$. By a generalized Fibonacci sequence $\{V_n{\}}_{n\ge 1}$ we mean any sequence of positive integers satisfying the recurrence $V_n=V_{n-1}+V_{n-2}$ for $n\ge 3$. We study the problem of determining the Frobenius number and genus for sets $A=\{V_n,V_{n+d},V_{n+2d},\dots \}$ for arbitrary $n$ and even $d$.

29 pages, 3 sections, 10 references. arXiv admin note: text overlap with arXiv:2411.04465