Padovan and Perrin Numbers as Sums of Two Jacobsthal Numbers

TL;DR

This paper investigates which Padovan and Perrin numbers can be expressed as the sum of two Jacobsthal numbers, providing a complete characterization of such cases.

Contribution

It determines all instances where Padovan and Perrin numbers are sums of two Jacobsthal numbers, a novel characterization in the study of these sequences.

Findings

Identified all Padovan numbers that are sums of two Jacobsthal numbers.

Identified all Perrin numbers that are sums of two Jacobsthal numbers.

Provided explicit solutions for the sum representations.

Abstract

Let $\left\lbrace P_{k}\right\rbrace_{k\geq0}$ be the Padovan sequence defined by $P_{k}=P_{k-2}+P_{k-3}$ with initial values are $P_{0}=P_{1}=P_{2}=1$. Let $\left\lbrace R_{k}\right\rbrace_{k\geq0}$ be the Perrin sequence defined by $R_{k}=R_{k-2}+R_{k-3}$ with initial values are $R_{0}=3$, $R_{1}=0$, $R_{2}=2$. And let $\left\lbrace J_{n}\right\rbrace_{n\geq0}$ be the Jacobsthal sequence defined by $J_n=2J_{n-1}+J_{n-2}$ with initials $J_0=0$, $J_1=1$. In this paper we determine all Padovan and Perrin numbers which are sum of two Jacobsthal number.