The size of $k$-th order generalized Fibonacci cubes

Abstract: Let $k\geq2$. Then the $k$-th order Fibonacci cube $Γ^{(k)}_{n}$ is the subgraph of the hypercube $Q_{n}$ induced by vertices without $k$ consecutive $1$s. The case $k=2$ corresponds to the classic Fibonacci cube $Γ{n}$. There are three kinds of calculation formulas of the size of $Γ{n}$: the iteration form $|E(Γ{n})|=|E(Γ{n-1})|+|E(Γ{n-2})|+F{n}$ (Hsu, 1993), %iteration form the convolution form $|E(Γ{n})|=\mathop{\sum}\limits{i=1}^{n}F_{i}F_{n-i+1}$ (Klavžar, 2005) %convolution form and the linear form $|E(Γ{n})|=\frac{nF{n+1}+2(n+1)F_{n}}{5}$ (Munarini et al., 2001). %linear form Belbachir and Ould-Mohamed (2020) studied the iteration and convolution formulas of the size of $Γ^{(3)}_{n}$. Very recently, Mollard (2025) deduced the iteration formula of the size of $Γ^{(k)}_{n}$ for $k\geq2$. In this paper, we give the the formulas of convolution and linear forms of $|E(Γ^{(k)}_{n})|$ for all $k\geq2$. Specifically, we obtain the formula of $|E(Γ^{(k)}_{n})|$ in terms of convolved $k$-th order Fibonacci numbers and the formula of $|E(Γ^{(k)}_{n})|$ of linear expression of $k$ consecutive $k$-th order Fibonacci numbers.