Abstract
This paper both extends and generalizes recently published properties which have been developed by many authors for elements of the Leonardo sequence in the context of second-order recursive sequences. It does this by considering the difference equation properties of the homogeneous Fibonacci sequence and the non-homogeneous properties of their Leonardo sequence counterparts. This produces a number of new identities associated with a generalized Leonardo sequence and its associated algorithm, as well as some combinatorial results which lead into elegant properties of hyper-Fibonacci numbers in contrast to their ordinary Fibonacci number analogues, and as a convolution of Fibonacci and Leonardo numbers.
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