### DIMACS - RUTGERS EXPERIMENTAL MATHEMATICS SEMINAR

Sponsored by the Rutgers University Department of Mathematics and the
Center for Discrete Mathematics and Theoretical Computer Science (DIMACS)

Co-organizers:
Drew Sills, Rutgers University, asills {at} math [dot] rutgers [dot] edu
Doron Zeilberger, Rutgers University, zeilberg {at} math [dot] rutgers [dot] edu

Title: Counting the Sums of Cubes of Fibonacci Numbers

Speaker: Arthur T. Benjamin, Harvey Mudd College

Date: October 7, 2004 4:30-5:30pm

Location: Hill Center, Room 425, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

We provide the first combinatorial proof for the sum of the cubes of the first n Fibonacci numbers. Specifically, we prove that $$\sum_{k=0}^n {f_k}^3 = (f_{3n+4} + (-1)^n 6 f_{n-1} + 5)/10,$$ where $f_n$ is the $n$th Fibonacci number defined by $f_0 =1$, $f_1=1$ and for $n \geq 2$, $f_n = f_{n-1} + f_{n-2}$. Along the way, elegant combinatorial proofs are also given for other Fibonacci identities. This is joint work with undergraduate Timothy Carnes.