On combinatorial properties of one type of Hessenberg matrix and the Leonardo numbers

In this paper, we introduced and investigated a Hessenberg matrix $T_n$, which incorporates both local interactions of tridiagonal matrices and fixed-distance non-local couplings, thereby exhibiting a hybrid Hessenberg–tridiagonal structure. We established a recurrence relation for its determinant and demonstrate that the determinant of $T_n$ coincides with the Leonardo numbers. As a consequence, alternative representations involving Fibonacci numbers and Chebyshev polynomials are also obtained. The characteristic polynomial of $T_n$ is derived through a Schur complement approach, and its spectral properties are examined in detail. The eigenvalue distribution reveals non-Hermitian features, including the presence of complex conjugate pairs and oscillatory instabilities. Moreover, an explicit LU factorization is obtained, enabling efficient computation of determinants and shedding light on the sparse structural properties of the matrix. Numerical experiments and visualization of eigenvalue behavior confirm the theoretical findings and highlight the potential of $T_n$ as a model for structured non-Hermitian operators relevant in dynamical systems, wave propagation, and transport phenomena.