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

TL;DR

This paper investigates the Frobenius number and genus for specific sets derived from generalized Fibonacci sequences, extending classical results to new structured sequences with particular index patterns.

Contribution

It provides formulas and methods for calculating Frobenius numbers and genus for sets formed from generalized Fibonacci sequences with indices in arithmetic progression.

Findings

Derived explicit formulas for Frobenius numbers.

Established methods for calculating the genus.

Extended classical Frobenius problem results to generalized Fibonacci sets.

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 $d=2$.