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Title: Determinant formulas of some Hessenberg matrices with Jacobsthal entries
Authors: Гой, Тарас Петрович
Shattuck, Mark
Keywords: Hessenberg matrix
Jacobsthal number
Jacobsthal-Lucas number
Mersenne number
Trudi formula
Issue Date: Jun-2021
Citation: Goy, T. Determinant formulas of some Hessenberg matrices with Jacobsthal entries / T. Goy, M. Shattuck // Applications and Applied Mathematics: An International Journal, 16(1) (2020), 191-213
Abstract: In this paper, we evaluate determinants of several families of Hessenberg matrices having various subsequences of the Jacobsthal sequence as their nonzero entries. These identities may be written equivalently as formulas for certain linearly recurrent sequences expressed in terms of sums of products of Jacobsthal numbers with multinomial coefficients. Among the sequences that arise in this way include the Mersenne, Lucas and Jacobsthal-Lucas numbers as well as the squares of the Jacobsthal and Mersenne sequences. These results are extended to Hessenberg determinants involving sequences that are derived from two general families of linear second-order recurrences. Finally, combinatorial proofs are provided for several of our determinant results which make use of various correspondences between Jacobsthal tilings and certain restricted classes of binary words.
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