A note on Pell-Padovan numbers and their connection with Fibonacci numbers

Abstract

In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

A comprehensive survey of dual-generalized complex Fibonacci and Lucas numbers

“…With a similar thought, our next goal is to present the DGC Oresme numbers which are given by the second-order (see in [43,44]) and examine the linear relations and, nonlinear properties of them, and their connection with DGC Fibonacci and Pell numbers. [38][39][40]…”

A Note on Neo Balancing Sequence, Generalized Recursive Sequences and Pythagorean Triples

“…From (10), we obtain the four consecutive terms as follows for 𝑡 = 1, is (2,5,4,0). From (10), we get the four consecutive terms as follows for 𝑡 = 2, which is (2,3,4,5). From (10), we get the four consecutive terms as follows…”

“…To explain the generation technique, theorems and demonstrations were supplied in-depth. 𝑝 2 + 𝑞 2 = 𝑟 2 (1) if and only if three positive integers 𝑝, 𝑞, 𝑟 form a Pythagorean triple. A right triangle with two smaller sides of length 𝑝, 𝑞 and a hypotenuse of length 𝑟 can be formed by any three positive integers that fulfill.…”

Oresme Hybrid Quaternion Numbers

“…Oresme numbers were generalized by Cook in [1]. On Oresme Numbers and their connection with Fibonacci and Pell Numbers by [4]. In [3], authors have given generalization of the matrix form of the Oresme sequence and Oresme's hybrid numbers.…”