Survey of Cubic Fibonacci Identities - When Cuboids Carry Weight

Abstract

The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter of mathematical curiosity, but also motivated by a quest for 3D Fibonacci spirals.As we were not able to find any survey on the particular topic of cubic Fibonacci identities we decided to try to fill this void. We started by surveying many Fibonacci identities and recording cubic ones. Obviously, tracing all Fibonacci identities (for identifying a handful) is a daunting task. Checking several hundred we have realized that it is not always clear who the author is. The reason is that in many cases an identity was stated in one article (sometimes without a proof, e.g., as an open problem, or a conjecture) while later being proven in another article, or effectively rediscovered independently by other authors. However, we have done our best to present the identities chronologically. We have supplied our own proof for one identity, having tried, but failed, to find a published proof. For all the other identities, we either proved them on a computer, or else verified by hand their original published proofs. Somehow unexpectedly, our investigations have revealed only a rather small number of cubic Fibonacci identities, representing a tiny fraction of all published Fibonacci identities (most of which are linear or quadratic). Finally, out of these, only a handful of cubic Fibonacci identities are homogeneous.

Cubic binomial Fibonacci sums

“…Nagy et al [4] noted that there is a dearth of cubic Fibonacci and Lucas identities in the existing literature. Some cubic Fibonacci identities with binomial coefficients were derived recently by Kronenburg [3].…”