Geonardo numbers

Abstract

Inspired by the term Gibonacci numbers, which was coined by A. T. Benjamin and J. J. Quinn as shorthand for generalized Fibonacci numbers, Geonardo numbers are considered. More concretely, for each $a\in\mathbb{N}_0$, the study of the sequence of generalized Leonardo numbers associated with $a$, introduced in an earlier work, is continued and new properties of these numbers are studied: parity; forms of Binet's formula; growth of consecutive Geonardo numbers plus $a \in \mathbb{N}_0$; generating functions -- ordinary, exponential, Poisson; identities -- sum-binomial, Catalan, Cassini, d'Ocagne, Melham. In addition, some unknown equalities and inequalities related to Leonardo numbers are previously established.