arxiv: 2604.20889 · v1 · submitted 2026-04-19 · 🧮 math.GM

Recognition: unknown

Structure and Growth of Galileo Sequences

William Cheah , David Treeby

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:08 UTC · model grok-4.3

classification 🧮 math.GM

keywords Galileo sequencepartial sumsdoubling propertypolynomial differencesbinary tree representationpower-law growthcontinuous analog

0 comments

The pith

Every polynomial Galileo sequence of positive integers must be the first difference a_n = C(n^d - (n-1)^d) for some constant C and degree d.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A Galileo sequence consists of positive integers whose partial sums S_n obey the fixed scaling S_{2n} = k S_n for some k > 1 and all n. The paper proves that any such sequence that is also a polynomial must arise exactly as the first differences of a scaled monomial, specifically a_n = C(n^d - (n-1)^d). It then establishes that every positive Galileo sequence admits a binary-tree representation of its terms. For the monotone integer-valued cases it derives power-law growth bounds and supplies a continuous version together with a full characterization of the continuous solutions.

Core claim

The central result is that a sequence of positive integers is a polynomial Galileo sequence if and only if it takes the explicit difference form a_n = C(n^d - (n-1)^d) for fixed positive integers C and d. This form automatically satisfies the doubling relation on partial sums with k = 2^d. The same scaling property is used to construct a binary-tree representation for arbitrary positive Galileo sequences and to obtain asymptotic power-law bounds when the sequence is additionally monotone.

What carries the argument

The Galileo doubling condition on partial sums S_{2n} = k S_n, which forces the sequence terms to be generated by a self-similar binary-tree structure or, in the polynomial case, by the exact first-difference formula a_n = C(n^d - (n-1)^d).

If this is right

The scaling constant must be a power of two: k = 2^d for the polynomial degree d.
Every positive Galileo sequence, polynomial or not, can be represented by a binary tree whose nodes encode the terms.
Monotone positive integer Galileo sequences obey power-law growth bounds on their partial sums.
The continuous analogs are fully characterized by the same scaling functional equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The binary-tree representation suggests that any Galileo sequence can be generated recursively by splitting intervals in a self-similar way, which may allow efficient computation or sampling.
The power-law bounds for monotone cases imply that the growth rate is determined solely by the doubling constant k, independent of other details of the sequence.
The continuous characterization may serve as a limit object for discrete sequences under suitable rescaling, opening the possibility of studying convergence rates.

Load-bearing premise

The sequence must consist of positive integers whose partial sums satisfy the exact doubling relation S_{2n} = k S_n for a fixed k greater than 1 and every positive integer n.

What would settle it

Exhibit a polynomial sequence of positive integers whose partial sums satisfy S_{2n} = k S_n for some fixed k > 1 but which cannot be written as C(n^d - (n-1)^d) for any constants C and d.

Figures

Figures reproduced from arXiv: 2604.20889 by David Treeby, William Cheah.

**Figure 1.** Figure 1: Equal children: a1 = 1, a2 = 3 given, then bn = cn = 2 for n ≥ 2. Example 7 (Unequal children for k = 4). For contrast, set bn = 1, cn = 3 for every n ≥ 2, so each parent splits as (1, 3). The resulting tree ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Unequal children: a1 = 1, a2 = 3 given, then (bn, cn) = (1, 3) for n ≥ 2. 5 Important examples As every Galileo sequence can be generated in this fashion, we can recover each of the families encountered thus far by some suitable choice of (bn, cn). Example 8 (Galileo’s sequence of odd integers). For the classical sequence an = 2n − 1 we have Sn = n 2 and hence k = 4. The splitting factors are bn = a2n−1 an… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper cleanly characterizes polynomial Galileo sequences from the scaling condition and adds a binary-tree view plus growth bounds.

read the letter

This paper defines Galileo sequences as positive integer sequences whose partial sums satisfy S_{2n} = k S_n for fixed k > 1. The main result is that every polynomial one must be a_n = C(n^d - (n-1)^d). That follows directly once you substitute a general polynomial into the functional equation: each lower-degree term scales by a different power of 2, so only the leading monomial survives. They then build a binary-tree representation that encodes the recursive doubling for any positive Galileo sequence, and for the monotone case they derive power-law growth bounds together with a continuous version that solves the analogous scaling equation for functions. The explicit polynomial form and the tree picture are the genuinely new pieces; they turn the abstract scaling rule into something concrete you can generate and draw. The reasoning for the polynomial case is tight and matches the stress-test observation that the equation forces the coefficients to vanish. The tree construction gives a useful combinatorial handle without introducing extra assumptions. The soft spots are modest. The continuous characterization is a straightforward extension of the same scaling relation and does not appear to reach beyond what is already known for homogeneous functional equations. The paper also stays inside its own definition and does not connect the sequences to existing families such as automatic sequences or self-similar sets, which keeps the scope narrow. No circularity or hidden fitting shows up in the claims. This is for people who work with recursive sequences, functional equations on integers, or discrete scaling structures. A reader already thinking about doubling maps or partial-sum constraints will find usable explicit examples and bounds. I would send it to referees; the core statements are specific enough to check and the supporting arguments look internally consistent.

Referee Report

0 major / 2 minor

Summary. The manuscript defines a Galileo sequence as a sequence of positive integers (a_n) whose partial sums S_n satisfy the scaling relation S_{2n} = k S_n for a fixed k > 1 and all positive integers n. It claims to prove that all such sequences that are polynomial are the first differences a_n = C (n^d - (n-1)^d) for some constant C and degree d. Additionally, it establishes a binary-tree representation for general positive Galileo sequences, proves power-law growth bounds for positive monotone integer-valued Galileo sequences, and characterizes all continuous solutions to the analogous scaling problem.

Significance. Should the proofs be correct, the paper offers a precise characterization of polynomial sequences satisfying the partial-sum scaling condition, showing they must be discrete derivatives of monomials. The binary-tree representation provides a combinatorial structure, and the growth bounds and continuous analog extend the results beyond polynomials. The derivation via substitution into the functional equation S_{2n} - k S_n = 0, forcing lower degree terms to vanish, is a standard and clean approach for such problems.

minor comments (2)

[Abstract] Abstract: the main polynomial result is stated without noting that k must equal 2^d; adding this relation would make the claim self-contained.
[Introduction] The definition of the partial sums S_n is used throughout but never written explicitly as S_n = sum_{i=1}^n a_i; a single displayed equation in the introduction would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of the results on Galileo sequences. The referee recommends minor revision, but the report contains no specific major comments or criticisms to address. We appreciate the concise summary of the paper's contributions and will incorporate any minor editorial improvements in the revised version.

Circularity Check

0 steps flagged

Derivation is self-contained algebraic consequence of the scaling condition

full rationale

The paper defines a Galileo sequence via the partial-sum condition S_{2n}=k S_n for fixed k>1 and positive integer terms. For the polynomial case, it assumes S_n is a polynomial of degree d and substitutes into the functional equation. Each monomial term n^j scales by the distinct factor 2^j when n is replaced by 2n, so the only way the equation holds identically is if all coefficients except the leading one vanish and k=2^d. The claimed form a_n=C(n^d-(n-1)^d) is then the immediate first difference of the resulting monomial. No parameter is fitted to data, no self-citation supplies the uniqueness, and no ansatz is smuggled in; the result follows directly from the definition and elementary polynomial algebra. The binary-tree representation and growth bounds are likewise presented as consequences of the same scaling without circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition of positive-integer sequences and the partial-sum scaling condition; no additional free parameters beyond the scaling constant C and degree d are introduced, and no new entities are postulated.

free parameters (2)

C
Arbitrary positive scaling constant appearing in the closed form for polynomial members.
d
Positive integer degree of the underlying polynomial.

axioms (2)

domain assumption The sequence consists of positive integers and the partial-sum relation S_{2n} = k S_n holds for a fixed k > 1 and all positive integers n.
This is the defining property of a Galileo sequence used throughout the proofs.
domain assumption For the polynomial case, a_n is a polynomial function of n.
The first theorem restricts to polynomial sequences.

pith-pipeline@v0.9.0 · 5381 in / 1508 out tokens · 52547 ms · 2026-05-10T06:08:40.944886+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references · 4 canonical work pages

[1]

Discrete Math

¨Omer E˘ gecio˘ glu.A note on iterated Galileo sequences. Discrete Math. Algorithms Appl. 9(2017), Article 1750015. DOI: 10.1142/S179383091750015X

work page doi:10.1142/s179383091750015x 2017
[2]

Translated by H

Galileo Galilei.Dialogues Concerning Two New Sciences. Translated by H. Crew and Alfonso de Salvio. Macmillan, New York, 1914. (Original work published 1638 asDiscorsi e Dimostrazioni Matematiche Intorno a Due Nuove Scienze.)

1914
[3]

May.Galileo sequences, a good dangling problem

Kenneth O. May.Galileo sequences, a good dangling problem. Amer. Math. Monthly79 (1972), 67–69. DOI: 10.2307/2978135

work page doi:10.2307/2978135 1972
[4]

Missouri J

Hari Pulapaka.On generalized alternating Galileo sequences. Missouri J. Math. Sci. 20(3) (2008), 178–185. DOI: 10.35834/mjms/1316032777

work page doi:10.35834/mjms/1316032777 2008
[5]

Tattersall.Elementary Number Theory in Nine Chapters, 2nd ed

James J. Tattersall.Elementary Number Theory in Nine Chapters, 2nd ed. Cambridge Univ. Press, Cambridge, 2005

2005
[6]

David Zeitlin.A family of Galileo sequences. Amer. Math. Monthly82(1975), 819–822. DOI: 10.2307/2319798

work page doi:10.2307/2319798 1975
[7]

Published elec- tronically athttps://oeis.org

OEIS Foundation Inc.The On-Line Encyclopedia of Integer Sequences. Published elec- tronically athttps://oeis.org. 12