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\begin{center}
    \vskip 1cm{\LARGE\bf  
    An Explicit Formulation for \\
    Supergeneralized Leonardo $p$-numbers \\}
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    \large
    Clemens Sch\"utz and Kristian Kelly \\ 
    Department of Mathematics\\
    University of Vienna\\
    Vienna \\
    Austria \\
    \href{mailto:schuetzc48@univie.ac.at }{\tt schuetzc48@univie.ac.at}\\
    \href{mailto:kristiank22@univie.ac.at}{\tt kristiank22@univie.ac.at}
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\begin{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 $n > p$.
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