The purpose of this article is to obtain some new infinite families of Toeplitz matrices, 7-matrices and generalized Pascal triangles whose leading principal minors form the Fibonacci, Lucas, Pell and Jacobsthal sequences. We also present a new proof for Theorem 3.1 in [R. Bacher. Determinants of matrices related to the Pascal triangle. J. Théor. Nombres Bordeaux, 14:19–41, 2002.].

We introduce and study an invertible transformation on integer sequences related to the Catalan numbers. Transformation pairs are identified among classical sequences. A closely related transformation which we call the generalized Ballot transform is also studied, along with associated transformations. Results concerning the Fibonacci, Jacobsthal and Pell numbers are derived. Finally, we derive...

In this paper, we define five parameters generalization of Fibonacci numbers that generalizes Fibonacci, Pell, Modified Jacobsthal, Narayana, Padovan, k-Fibonacci, k-Pell, k-Jacobsthal and p-numbers, distance numbers, (2, k)-distance generalized (k, r)-Fibonacci in the sense by extending definition a recurrence relation with two adding three distance, simultaneously. Tiling combinatorial interp...