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

Abstract

For a set $A$ of positive integers with $\gcd(A)=1$, let $\langle A \rangle$ denote the set of all finite linear combinations of elements of $A$ over the non-negative integers. The it is well known that only finitely many positive integers do not belong to $\langle A \rangle$. 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 $A=\{V_n,V_{n+d},V_{n+2d},\ldots\}$ for arbitrary $n$, where $d$ odd or $d=2$.

Theorems & Definitions (18)

• Proposition 1.1
• Proposition 2.1
• Corollary 2.2
• Corollary 2.3
• Theorem 2.4
• Theorem 3.1
• Theorem 3.2
• Proposition 4.1
• Lemma 4.3
• Theorem 4.4