  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 an r-th order recurrence of a more
general form with the appropriate initial conditions, where r>2 is arbitrary.
