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An Explicit Formulation for
Supergeneralized Leonardo $p$-numbers
Clemens Schütz and Kristian Kelly
Department of Mathematics
University of Vienna
Vienna
Austria
schuetzc48@univie.ac.at
kristiank22@univie.ac.at

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Abstract:

In this paper, we introduce the supergeneralized Leonardo $p$-numbers, $\mathcal{L}_{p,k,\mathbf{x}}(n)$, which extend the definition of the generalized Leonardo $p$-numbers, introduced by Kuhapatanakul and Ruankong, by not requiring $\mathcal{L}_{p,k}(0) = \cdots = \mathcal{L}_{p,k}(p) = 1$ but allowing the first $p+1$ initial values to be chosen freely. We then investigate the structure of these sequences, show that they are related to the Fibonacci $p$-numbers and provide an explicit formulation for $\mathcal{L}_{p,k,\mathbf{x}}(n)$ when $n > p$.