Binomial transform of particular sequence, it’s some identities and summation formulas

In this paper we introduce a new particular number sequence, namely the Jacobsthal-Leonardo sequence, which is a combination of the Jacobsthal sequence and Leonardo sequence. We give the Binet-like formula, partial sum, generating function and exponential generating function related to this sequence. Also, Cassini’s identity, other interesting identities and summation formulas of this sequence are stated in this paper. Furthermore, we apply the binomial transform to this sequence and investigate some identities and give some summation formulas of it. In order to highlight the results, some numerical examples are given in this paper. Finally, we give a PYTHON code to generate the first 𝑛 terms of the Jacobsthal-Leonardo sequence.

Keywords:

Jacobsthal sequence , Leonardo number , Binet Formula , Summation formula , Binomial transform