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

Simon R. Cowell, Mariana Nagy, Valeriu Beiu


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).


Combinatorics, Fibonacci sequence, Identities

Full Text:



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.


  • There are currently no refbacks.