Some Identities for Palindromic Compositions Without $2$'s
Received:March 03, 2017  Revised:May 24, 2017
Key Word: palindrome   the Fibonacci number   identity   combinatorial proof  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11461020).
Author NameAffiliation
Yuhong GUO School of Mathematics and Statistics, Hexi University, Gansu 734000, P. R. China 
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      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 $1$.
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