Geodesic
The 99th Fibonacci identity
The electronic journal of combinatorics, Tome 15 (2008)
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We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences $G_n$ that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well. Among these, we prove $\sum_{k\geq 0}{n \choose k}G_{2k} = 5^n G_{2n}$ and $\sum_{k\geq 0}{n \choose k}G_{qk}(F_{q-2})^{n-k} = (F_q)^n G_{2n}$.
DOI : 10.37236/758
Classification : 05A19, 11B39
Mots-clés : Fibonacci number identities, Lucas number identities, Fibonacci sequences, Gibonacci sequences 
@article{10_37236_758,
     author = {Arthur T. Benjamin and Alex K. Eustis and Sean S. Plott},
     title = {The 99th {Fibonacci} identity},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/758},
     zbl = {1158.05305},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/758/}
}
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%A Sean S. Plott
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Arthur T. Benjamin; Alex K. Eustis; Sean S. Plott. The 99th Fibonacci identity. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/758

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