Hyperfibonacci Sequences and Polytopic Numbers

TL;DR

This paper explores properties of hyperfibonacci sequences, establishing their relation to polytopic numbers, providing recursive definitions, extending Binet formulas, and deriving identities involving hyperfibonacci and hyperlucas sequences.

Contribution

It introduces new identities and recursive formulas for hyperfibonacci sequences and their connections to polytopic numbers, extending classical Fibonacci analysis.

Findings

Difference between hyperfibonacci and predecessors equals polytopic number

Derived recursive definitions for hyperfibonacci sequences

Extended Binet formula for hyperfibonacci sequences

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.