Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University
Department of Mathematics
University of Tennessee
Knoxville, TN 37996
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.
Full version: pdf,
(Concerned with sequences
Received December 9 2019; revised version received April 2 2020.
Published in Journal of Integer Sequences,
June 11 2020.
Journal of Integer Sequences home page