In this paper, we introduce the supergeneralized Leonardo <SPAN CLASS="MATH"><IMG
 STYLE="height: 1.64ex; vertical-align: -0.55ex; " SRC="img2.svg"
 ALT="$p$"></SPAN>-numbers, <!-- MATH
 $\mathcal{L}_{p,k,\mathbf{x}}(n)$
 -->
<SPAN CLASS="MATH"><IMG
 STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="img3.svg"
 ALT="$\mathcal{L}_{p,k,\mathbf{x}}(n)$"></SPAN>, which extend the definition of the generalized Leonardo <SPAN CLASS="MATH"><IMG
 STYLE="height: 1.64ex; vertical-align: -0.55ex; " SRC="img2.svg"
 ALT="$p$"></SPAN>-numbers, introduced by Kuhapatanakul and Ruankong, by not requiring <!-- MATH
 $\mathcal{L}_{p,k}(0) = \cdots = \mathcal{L}_{p,k}(p) = 1$
 -->
<SPAN CLASS="MATH"><IMG
 STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="img4.svg"
 ALT="$\mathcal{L}_{p,k}(0) = \cdots = \mathcal{L}_{p,k}(p) = 1$"></SPAN> but allowing the first <SPAN CLASS="MATH"><IMG
 STYLE="height: 2.13ex; vertical-align: -0.55ex; " SRC="img5.svg"
 ALT="$p+1$"></SPAN> initial values to be chosen freely.
We then investigate the structure of these sequences, show that they are related to the Fibonacci <SPAN CLASS="MATH"><IMG
 STYLE="height: 1.64ex; vertical-align: -0.55ex; " SRC="img2.svg"
 ALT="$p$"></SPAN>-numbers and provide an explicit formulation for <!-- MATH
 $\mathcal{L}_{p,k,\mathbf{x}}(n)$
 -->
<SPAN CLASS="MATH"><IMG
 STYLE="height: 2.58ex; vertical-align: -0.75ex; " SRC="img3.svg"
 ALT="$\mathcal{L}_{p,k,\mathbf{x}}(n)$"></SPAN> when <SPAN CLASS="MATH"><IMG
 STYLE="height: 1.86ex; vertical-align: -0.55ex; " SRC="img6.svg"
 ALT="$n &gt; p$"></SPAN>.