Sum of Consecutive Terms of Pell and Related Sequences

TL;DR

This paper investigates identities involving sums of consecutive Pell and related sequences, establishing conditions under which these sums are multiples of sequence terms, and extends results to Fibonacci and similar sequences.

Contribution

It introduces new identities for sums of consecutive Pell numbers and their generalizations, characterizes when these sums are multiples of sequence terms, and extends findings to other recursive sequences.

Findings

Sum of N>1 Pell numbers is a multiple of another Pell number iff 4 divides N.

Sum of 2k+2 consecutive generalized Pell numbers is a multiple of another sequence term.

Certain relations do not hold for odd N and k in generalized Pell sequences.

Abstract

We study new identities related to the sums of adjacent terms in the Pell sequence, defined by $P_{n} := 2P_{n-1}+P_{n-2}$ for $ n\geq 2$ and $P_{0}=0, P_{1}=1$, and generalize these identities for many similar sequences. We prove that the sum of $N>1$ consecutive Pell numbers is a fixed integer multiple of another Pell number if and only if $4\mid N$. We consider the generalized Pell $(k,i)$-numbers defined by $p(n) :=\ 2p(n-1)+p(n-k-1) $ for $n\geq k+1$, with $p(0)=p(1)=\cdots =p(i)=0$ and $p(i+1)=\cdots = p(k)=1$ for $0\leq i\leq k-1$, and prove that the sum of $N=2k+2$ consecutive terms is a fixed integer multiple of another term in the sequence. We also prove that for the generalized Pell $(k,k-1)$-numbers such a relation does not exist when $N$ and $k$ are odd. We give analogous results for the Fibonacci and other related second-order recursive sequences.