Pell and associated Pell braid sequences as GCDs of sums of $k$ consecutive Pell, balancing, and related numbers

TL;DR

This paper investigates the GCDs of sums of $k$ consecutive terms from six well-known sequences, providing explicit formulas and revealing their connection to Pell braid sequences, with additional insights on squared sums and open problems.

Contribution

It offers explicit closed-form formulas for GCDs of sums of consecutive terms across six sequences and links these GCDs to Pell braid sequences, advancing understanding of their algebraic structure.

Findings

Closed-form formulas for GCDs of sums of $k$ consecutive terms.

Representation of GCDs as Pell braid sequences.

Partial results and open questions on squared sums.

Abstract

We consider the greatest common divisor (GCD) of all sums of $k$ consecutive terms of a sequence $(S_n)_{n\geq 0}$ where the terms $S_n$ come from exactly one of following six well-known sequences' terms: Pell $P_n$, associated Pell $Q_n$, balancing $B_n$, Lucas-balancing $C_n$, cobalancing $b_n$, and Lucas-cobalancing $c_n$ numbers. For each of the six GCDs, we provide closed forms dependent on $k$. Moreover, each of these closed forms can be realized as braid sequences of Pell and associated Pell numbers in an intriguing manner. We end with partial results on GCDs of sums of squared terms and open questions.