Sum of Consecutive Terms of Pell and Related Sequences

TL;DR

This work investigates when sums of consecutive terms in Pell-type sequences are fixed integer multiples of other terms. Using generalized Binet formulas, generating-function techniques, and tiling interpretations, it establishes that for Pell numbers the sum of $4N$ consecutive terms equals a constant times $P(n+2N)$, with the constant given by $(a^{2N}-b^{2N})/\sqrt{2}$, and shows that similar fixed-multiple relations fail for sums of lengths $4N+2$ or many odd-length cases. The results extend to generalized Pell $(k,i)$-numbers and to Fibonacci and Lucas sequences, including explicit identities like $ extstyle\sum_{i=0}^{4N+1} F(n+i)=L(2N+1)F(n+2N+2)$ and nonexistence statements for several other lengths. A general Lucas-sequence framework is developed to delineate when such fixed-multiple sums can occur, and conjectures are offered for the generalized Pell family, supported by tiling interpretations and computational experiments. Overall, the paper advances the understanding of additive structure in second-order recurrences and links algebraic, combinatorial, and computational perspectives.

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.

Theorems & Definitions (37)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Example
• Definition 1.5
• Theorem 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3