Journal of Integer Sequences, Vol. 23 (2020), Article 20.6.8

Some Toeplitz-Hessenberg Determinant Identities for the Tetranacci Numbers

Taras Goy
Faculty of Mathematics and Computer Science
Vasyl Stefanyk Precarpathian National University

Mark Shattuck
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.

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(Concerned with sequences A000045 A000073 A000078 A001591 A001592 A079262 A104144 A122189.)

Received December 9 2019; revised version received April 2 2020. Published in Journal of Integer Sequences, June 11 2020.

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