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**Unformatted text preview: **INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 6 (2006), #A02 A BIJECTIVE PROOF OF f n +4 + f 1 + 2 f 2 + Â· Â· Â· + nf n = ( n + 1) f n +2 + 3 Philip Matchett Wood Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, USA [email protected] Received: 10/6/05, Revised: 1/5/06, Accepted: 1/16/06, Published: 2/1/06 Abstract In Proofs that Really Count , Benjamin and Quinn mentioned that there was no known bijective proof for the identity f 1 + 2 f 2 + Â· Â· Â· + nf n = ( n + 1) f n +2- f n +4 + 3 for n â‰¥ 0, where f k is the k-th Fibonacci number. In this paper, we interpret f k as the cardinality of the set F k consisting of all ordered lists of 1â€™s and 2â€™s whose sum is k . We then demonstrate a bijection between the sets F n +4 âˆª n k =1 ( { 1 , 2 , . . . , k } Ã— F k ) and ( { 1 , 2 , . . . , n + 1 } Ã— F n +2 ) âˆª { 1 , 2 , 3 } , which gives a bijective proof of the identity. 1. Introduction We will interpret the k-th Fibonacci number f k as the cardinality of the set F k of all ordered lists of 1â€™s and 2â€™s that have sum k . Thus, ( f , f 1 , f 2 , f 3 , f 4 , f 5 , . . . ) = (1 , 1 , 2 , 3 , 5 , 8 , . . . ). For an integer m , the number mf k will be interpreted as the cardinality of the Cartesian product [ m ] Ã— F k , where [ m ] := { 1 , 2 , 3 , . . . , m } . On page 14 of Proofs that Really Count [1], Benjamin and Quinn mentioned that there was no known bijective proof for the identity f 1 + 2 f 2 + . . . + nf n = ( n + 1) f n +2- f n +4 + 3 for n â‰¥ 0. In Section 2 we define a map Ï† : F n +4 âˆª n k =1 ([ k ] Ã— F k )-â†’ { 1 , 2 , 3 } âˆª ([ n + 1] Ã— F n +2 ) , and in Section 3 we describe why...

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