Some Toeplitz-Hessenberg determinant identities for the tetranacci numbers

Abstract

In this paper, we consider families of Toeplitz-Hessenberg determinants the entries of which are tetranacci numbers. In several cases, it is found that these determinants have simple closed form expressions in terms of well-known combinatorial sequences. Equivalently, the determinant formulas may be expressed as identities involving sums of products of tetranacci numbers and multinomial coefficients. In particular, we establish a connection between the tetranacci and both the Fibonacci and tribonacci number sequences via Toeplitz-Hessenberg determinants. Finally, combinatorial proofs that make use of sign-changing involutions and the formal definition of the determinant as a signed sum over the permutation group may be provided for several of the identities.