We provide some interesting relations involving $k$-generalized Fibonacci numbers between the set $F_n^{(k)}$ of length $n$ binary strings avoiding $k$ of consecutive $0$'s and the set of length $n$ strings avoiding $k+1$ consecutive $0$'s and $1$'s with some more restriction on the first and last letter, via a simple bijection. In the special case $k=2$ a probably new interpretation of Fibonacci numbers is given. Moreover, we describe in a combinatorial way the relation between the strings of $F_n^{(k)}$ with an odd numbers of $1$'s and the ones with an even number of $1$'s.
Restricted binary strings and generalized Fibonacci numbers / Bernini, Antonio. - STAMPA. - 10248:(2017), pp. 32-43. (Intervento presentato al convegno CELLULAR AUTOMATA THEORY AND APPLICATINS tenutosi a Milano nel 7-9 Giugno 2017) [10.1007/978-3-319-58631-1_3].
Restricted binary strings and generalized Fibonacci numbers
BERNINI, ANTONIO
2017
Abstract
We provide some interesting relations involving $k$-generalized Fibonacci numbers between the set $F_n^{(k)}$ of length $n$ binary strings avoiding $k$ of consecutive $0$'s and the set of length $n$ strings avoiding $k+1$ consecutive $0$'s and $1$'s with some more restriction on the first and last letter, via a simple bijection. In the special case $k=2$ a probably new interpretation of Fibonacci numbers is given. Moreover, we describe in a combinatorial way the relation between the strings of $F_n^{(k)}$ with an odd numbers of $1$'s and the ones with an even number of $1$'s.File | Dimensione | Formato | |
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