Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.7

Determinant of Three-Layer Toeplitz Matrices

Dmitry Efimov
Institute of Physics and Mathematics
Komi Science Centre UrD RAS
Kommunisticheskaya Street 24
167000 Syktyvkar


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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