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

TL;DR

This paper investigates the Frobenius number and genus for sets derived from generalized Fibonacci sequences, specifically focusing on sequences with indices spaced evenly, extending classical results to these structured sets.

Contribution

It provides new formulas and insights for the Frobenius number and genus for sets of generalized Fibonacci numbers with evenly spaced indices.

Findings

Derived explicit formulas for Frobenius numbers.

Extended classical Frobenius problem results to generalized Fibonacci sets.

Analyzed cases with even index spacing in Fibonacci sequences.

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. Then 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 $n$ and even $d$.