Hyperfibonacci Sequences and Polytopic Numbers

Abstract

We prove that the difference between the $n$-th hyperfibonacci number of $r$-th generation and its two consecutive predecessors is the $n$-th regular $(r-1)$-topic number. Using this fact we provide an equivalent recursive definition of hyperfibonacci sequences and derive an extension of the Binet formula. We also prove further identities involving both hyperfibonacci and hyperlucas sequences, in full generality.