A Proof of a Generic Fibonacci Identity From Wolfram’s MathWorld

Simon R. Cowell, Mariana Nagy, Valeriu Beiu

Abstract


The generic identity $F_{kn+c}=\sum_{i=0}^k C_k^i F_{c-i} F_n^i F_{n+1}^{k-i}$ involving the Fibonacci numbers $F_n$ (where $C_k^i$ denotes the binomial coefficient counting the number of choices of $i$ elements from a set of $k$ elements), is attributed on Wolfram's MathWorld website  (Chandra & Weisstein, 2018) to a personal communication from Aleksandrs Mihailovs. In spite of a very thorough search, we have been unable to find a published proof. We present here a combinatorial proof of our own, using the methods of Benjamin, Eustis and Plott (Benjamin et al., 2008).


Keywords


Combinatorics, Fibonacci sequence, Identities

Full Text:

PDF

References


Benjamin, A. T., A. K. Eustis and S. S. Plott (2008). The 99th Fibonacci identity. Electr. J. Combinatorics 15(R34), 1– 13.

Chandra, P. and E. W. Weisstein (2018). “Fibonacci Number”. From MathWorld, A Wolfram Web Resource. http:

//mathworld.wolfram.com/FibonacciNumber.html. Accessed 22-04-2018.

Sloane, N. J. A. (1964). Online encyclopedia of integer sequences (OEIS), A000045. https://oeis.org/A000045. Accessed 22-04-2018.


Refbacks

  • There are currently no refbacks.