Determinants of Toeplitz–Hessenberg Matrices with Generalized Leonardo Number Entries

Let u_n = u_n(k) denote the generalized Leonardo number defined recursively by u_n = u_n−1 + u_n−2 + k for n ≥ 2, where u₀ = u₁ = 1. Terms of the sequence u_n(1) are referred to simply as Leonardo numbers. In this paper, we find expressions for the determinants of several Toeplitz–Hessenberg matrices having generalized Leonardo number entries. These results are obtained as special cases of more general formulas for the generating function of the corresponding sequence of determinants. Special attention is paid to the cases 1 ≤ k ≤ 7, where several connections are made to entries in the On-Line Encyclopedia of Integer Sequences. By Trudi’s formula, one obtains equivalent multi-sum identities involving sums of products of generalized Leonardo numbers. Finally, in the case k = 1, we also provide combinatorial proofs of the determinant formulas, where we make extensive use of sign-changing involutions on the related structures.