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Permutation Count Modulo Four

2175ExpertCombinatoricsCounting Principles

二〇一五數學奧林匹亞競賽第三階段選訓營

Consider the permutation of $1, 2, \dots, n$, which we denote as $\{a_1, a_2, \dots, a_n\}$. Let $f(n)$ be the number of these permutations satisfying the following conditions: (1) $a_1 = 1$; (2) $|a_i - a_{i-1}| \le 2, i = 1, 2, \dots, n-1$. What is the residue when we divide $f(2015)$ by 4?
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