Applications of Fibonacci Numbers pp 25-28 | Cite as
Recounting Binomial Fibonacci Identities
Conference paper
Abstract
In [4], Carlitz demonstrates using sophisticated matrix methods and Binet’s formula. Nevertheless, the presence of binomial coefficients suggests that an elementary combinatorial proof should be possible. In this paper, we present such a proof, leading to other Fibonacci identities.
$$ {F_L}\sum\limits_{{x_1} = 0}^n {\sum\limits_{{x_2} = 0}^n {...\sum\limits_{{x_{L = 0}}}^n {\left( \begin{gathered}n - {x_L} \hfill \\{x_1} \hfill \\\end{gathered} \right)} } } \left( \begin{gathered}n - {x_1} \hfill \\{x_2} \hfill \\\end{gathered} \right)...\left( \begin{gathered}n - {x_L} - 1 \hfill \\{x_L} \hfill \\\end{gathered} \right) = {F_{\left( {n + 1} \right)L,}} $$
(1)
AMS Classification Numbers
05A19 11B39Preview
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References
- [1]Benjamin, A.T. and Quinn, J.J. “Recounting Fibonacci and Lucas Identities.” College Mathematics Journal, Vol. 30.5 (1999): pp. 359–366.MathSciNetMATHCrossRefGoogle Scholar
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- [3]Brigham, R.C., Caron, R.M., Chinn, P.Z. and Grimaldi, R.P. “A Tiling Scheme for the Fibonacci Numbers.” J. Recreational Math, Vol. 28.1 (1996–97): pp. 10–16.Google Scholar
- [4]Carlitz, L. “The Characteristic Polynomial of a Certain Matrix of Binomial Coefficients.” The Fibonacci Quarterly, Vol. 3.2 (1965): pp. 81–89.MathSciNetMATHGoogle Scholar
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