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

Let $k\geq2$. Then the $k$-th order Fibonacci cube $\Gamma^{(k)}_{n}$ 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 $\Gamma_{n}$. There are three kinds of calculation formulas of the size of $\Gamma_{n}$: the iteration form $|E(\Gamma_{n})|=|E(\Gamma_{n-1})|+|E(\Gamma_{n-2})|+F_{n}$ (Hsu, 1993), %iteration form the convolution form $|E(\Gamma_{n})|=\mathop{\sum}\limits_{i=1}^{n}F_{i}F_{n-i+1}$ (Klav\v{z}ar, 2005) %convolution form and the linear form $|E(\Gamma_{n})|=\frac{nF_{n+1}+2(n+1)F_{n}}{5}$ (Munarini et al., 2001). %linear form Belbachir and Ould-Mohamed (2020) studied the iteration and convolution formulas of the size of $\Gamma^{(3)}_{n}$. Very recently, Mollard (2025) deduced the iteration formula of the size of $\Gamma^{(k)}_{n}$ for $k\geq2$. In this paper, we give the the formulas of convolution and linear forms of $|E(\Gamma^{(k)}_{n})|$ for all $k\geq2$. Specifically, we obtain the formula of $|E(\Gamma^{(k)}_{n})|$ in terms of convolved $k$-th order Fibonacci numbers and the formula of $|E(\Gamma^{(k)}_{n})|$ of linear expression of $k$ consecutive $k$-th order Fibonacci numbers.