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.