The Stern diatomic sequence via generalized Chebyshev polynomials

TL;DR

This paper links the Stern diatomic sequence to generalized Chebyshev polynomials and matrix determinants, providing new formulas and insights into its structure and computation.

Contribution

It introduces a novel representation of the Stern sequence using generalized Chebyshev polynomials and matrix determinants, expanding understanding of its algebraic properties.

Findings

A formula expressing a(n) via generalized Chebyshev polynomials.

Representation of a(n) as a matrix determinant involving distances between 1's in binary.

Connection between the sequence and polynomial evaluations with specific variable assignments.

Abstract

Let a(n) be the Stern's diatomic sequence, and let x1,...,xr be the distances between successive 1's in the binary expansion of the (odd) positive integer n. We show that a(n) is obtained by evaluating generalized Chebyshev polynomials when the variables are given the values x1+1, ..., xr+1, and we derive a formula expressing the same polynomials in terms of sets of increasing integers of alternating parity. We also show that a(n) = Det(Ir + Mr), where Ir is the rxr identity matrix, and Mr is the rxr matrix that has x1,...,xr along the main diagonal, then all 1's just above and below the main diagonal, and all the other entries are 0.