# Some Linear Recurrences Motivated by Stern's Diatomic Array

## TL;DR

This paper introduces a new triangular array related to Stern's diatomic array, demonstrating that sums of powers of its entries follow linear recurrences with potential for explicit formulas and broad generalizations.

## Contribution

It defines a novel array linked to Stern's array and proves that sums of powers satisfy linear recurrences, offering explicit cases and extensive generalizations.

## Key findings

- Sum of r-th powers in the array satisfy linear recurrences
- Explicit formulas obtained in special cases
- Generalization of recurrence relations

## Abstract

We define a triangular array closely related to Stern's diatomic array and show that for a fixed integer $r\geq 1$, the sum $u_r(n)$ of the $r$th powers of the entries in row $n$ satisfy a linear recurrence with constant coefficients. The proof technique yields a vast generalization. In certain cases we can be more explicit about the resulting linear recurrence.