Hyperfibonacci sequences and polytopic numbers

We prove that the difference between the $n$th hyperfibonacci number of the $r$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 the hyperfibonacci sequences, and derive an extension of the Binet formula. We also prove further identities involving both hyperfibonacci and hyperlucas sequences, in full generality.