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Determinant of Three-Layer Toeplitz Matrices
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Dmitry Efimov

Institute of Physics and Mathematics

Komi Science Centre UrD RAS

Kommunisticheskaya Street 24

167000 Syktyvkar

Russia

**Abstract:**

We obtain an explicit and efficient formula for the determinant of a
three-layer Toeplitz matrix. We show that many well-known sequences, such
as Jacobsthal numbers, generalized Fibonacci numbers, and *k*-Fibonacci
numbers, can be represented as sequences of determinants of three-layer
Toeplitz matrices. Further, we evaluate the spectrum for one of these
matrices using the obtained formulae and, as a consequence, discover
some interesting factorizations of certain integer sequences in terms of
products of complex numbers, the imaginary parts of which are expressed
using the tangent function.

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(Concerned with sequences
A000045
A000129
A001045
A003665
A007051
A015441
A046717
A054878
A078008
A102901
A320469.)

Received June 10 2021; revised version received August 20 2021.
Published in *Journal of Integer Sequences*,
October 25 2021.

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