q-cube polynomial of Tribonacci cube

Tribonacci cubes, also known as third generalized Fibonacci cubes, are subgraphs of the n-cube induced by the set of binary strings that do not contain three consecutive $1$'s. We investigate the number of hypercubes at a certain distance from the null-weight vertex in Tribonacci cubes, where the weight of a vertex is determined by the number of $1$'s it contains. The counting polynomial of these cubes is called the q-cube polynomial. We provide a recurrence relation for the q-cube polynomial of Tribonacci cubes. Additionally, we derive a formula for the k-th coefficient of these polynomials, expressed in terms of the Tricobsthal polynomial. Finally, we determine the generating function for these coefficients.