Structure and Growth of Galileo Sequences

TL;DR

This paper characterizes Galileo sequences, showing polynomial sequences are differences of specific forms, representing all positive sequences with binary trees, and establishing growth bounds for monotone sequences.

Contribution

It provides a complete characterization of polynomial Galileo sequences, a binary-tree representation for all positive sequences, and growth bounds for monotone sequences.

Findings

Polynomial Galileo sequences are differences of the form C(n^d - (n-1)^d)

Every positive Galileo sequence has a binary-tree representation

Monotone sequences exhibit power-law growth bounds

Abstract

A Galileo sequence \((a_n)\) is a sequence of positive integers whose partial sums $S_n$ satisfy $S_{2n}=kS_n$ for some $k>1$. In this paper we prove that every polynomial Galileo sequence is given by first differences of the form \(a_n= C\left(n^d-(n-1)^d\right)\). We then show that every positive Galileo sequence has a binary-tree representation. Finally, for positive monotone integer-valued Galileo sequences, we prove power-law growth bounds, and give a continuous analog together with a characterization of all continuous solutions.