Some Identities for Palindromic Compositions Without $2$'s |
Received:March 03, 2017 Revised:May 24, 2017 |
Key Words:
palindrome the Fibonacci number identity combinatorial proof
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Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11461020). |
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Abstract: |
In this paper, we study the palindromic compositions of even integers when no $2$'s are allowed in a composition and its conjugate. We show that the number of these palindromes is equal to $2F_{n-1}$, where, $F_n$ is the $n$-th Fibonacci number. Consequently, we obtain several identities between the number of these palindromes, the number of compositions into parts equal to $1$'s or $2$'s, the number of compositions into odd parts and the number of compositions into parts greater than $1$. |
Citation: |
DOI:10.3770/j.issn:2095-2651.2018.02.003 |
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