Identities for full-history Horadam sequences

TL;DR

This paper introduces a master combinatorial identity for full-history linear recurrences, enabling derivation of numerous identities for sequences like Pell, Jacobsthal, and m-nacci numbers.

Contribution

It presents a new combinatorial proof technique and a unifying identity for full-history linear recurrence sequences, including many known and new identities.

Findings

Derived a master identity for full-history sequences

Provided combinatorial proofs for multiple identities

Applied results to Pell, Jacobsthal, and m-nacci sequences

Abstract

We prove a master identity for a class of sequences defined by full-history linear homogeneous recurrences with (non-negative) constant coefficients. The identity is derived in a combinatorial way, providing thus combinatorial proofs for many known and new identities obtained as its corollaries. In particular, we prove several interesting identities for the Pell, the Jacobsthal, and the m-nacci numbers.