On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II
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| Format: | Preprint |
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2025
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| _version_ | 1866908428938510336 |
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| author | Shaikh, Ryan Azim Tripathi, Amitabha |
| author_facet | Shaikh, Ryan Azim Tripathi, Amitabha |
| contents | 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 $d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_00495 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II Shaikh, Ryan Azim Tripathi, Amitabha Number Theory 11D07, 20M14, 20M30 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 $d$. |
| title | On the Frobenius Problem for Some Generalized Fibonacci Subsequences -- II |
| topic | Number Theory 11D07, 20M14, 20M30 |
| url | https://arxiv.org/abs/2507.00495 |