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Express a_n in terms of Fibonacci numbers

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September 1, 2010
combinatoricsFibonacciFibonacci sequenceCombinatorics of wordsSequenceIMO ShortlistIMO Longlist

Problem Statement

Fibonacci numbers are defined as follows: F0=F1=1,Fn+2=Fn+1+Fn,n≥0F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0. Let ana_n be the number of words that consist of nn letters 00 or 11 and contain no two letters 11 at distance two from each other. Express ana_n in terms of Fibonacci numbers.
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