Determinant identities for toeplitz-hessenberg matrices with tribonacci entries

Abstract

In this paper‎, ‎we evaluate determinants of some families of Toeplitz--Hessenberg matrices having tribonacci number entries‎. ‎These determinant formulas may also be expressed equivalently as identities that involve sums of products of multinomial coefficients and tribonacci numbers‎. ‎In particular‎, ‎we establish a connection between the tribonacci and the Fibonacci and Padovan sequences via Toeplitz--Hessenberg determinants‎. ‎We then obtain‎, ‎by combinatorial arguments‎, ‎extensions of our determinant formulas in terms of generalized tribonacci sequences satisfying a recurrence of the form $T_n^{(r)}=T_{n-1}^{(r)}+T_{n-2}^{(r)}+T_{n-r}^{(r)}$ for $n geq r$‎, ‎with the appropriate initial conditions‎, ‎where $r geq 3$ is arbitrary‎.