University of IsfahanTransactions on Combinatorics2251-86579220200601Determinant identities for Toeplitz-Hessenberg matrices with tribonacci entries891092443710.22108/toc.2020.116257.1631ENTarasGoyFaculty of Mathematics and Computer Sciences, Vasyl Stefanyk Precarpathian National University, 76018, Ivano-Frankivsk, Ukraine0000-0002-6212-3095MarkShattuckDepartment of Mathematics
University of Tennessee
Knoxville, TN, 37996-1300Journal Article20190401In 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.https://toc.ui.ac.ir/article_24437_f2afe40be6184ea9d4bd30cc213590e8.pdf