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arxiv: 2309.01644 · v5 · pith:JGQR3YXCnew · submitted 2023-09-04 · 🧮 math.CO

The Pell Tower and Ostronometry

Robbert Fokkink This is my paper

Pith reviewed 2026-05-24 06:37 UTC · model grok-4.3

classification 🧮 math.CO
keywords bi-infinite sequenceslinear recurrencesPell sequencesnumeration systemscombinatorial patternsFibonacci generalizationsRed Wall
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The pith

Extending bi-infinite sequence tables from the Fibonacci recurrence to X_{n+1}=dX_n + X_{n-1} for natural d produces a Red Wall and exotic numeration systems.

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

The paper takes tables of bi-infinite sequences that Conway and Ryba built for Fibonacci numbers and rebuilds them using the more general recurrence with coefficient d for any natural number. It searches the new tables for patterns and identifies a Red Wall as one prominent feature along with links to unusual number representations. A sympathetic reader would care because the change in recurrence coefficient alters the structures that appear while still producing recognizable combinatorial objects. This shows that the original patterns are special cases within a larger family rather than isolated phenomena. If the extension holds, it supplies concrete new examples of how linear recurrences organize into visual and representational systems.

Core claim

By arranging bi-infinite sequences that obey the recurrence X_{n+1}=dX_n + X_{n-1} into tables for each natural number d, the author observes new patterns that include a Red Wall and that give rise to exotic numeration systems, thereby extending the earlier Fibonacci-table constructions.

What carries the argument

Bi-infinite tables of sequences obeying the recurrence X_{n+1}=dX_n + X_{n-1} for natural d, which generate the Red Wall boundary and the exotic numeration systems.

If this is right

  • The Fibonacci patterns become special cases inside a one-parameter family of tables.
  • A Red Wall appears as a distinct structural feature for d greater than 1.
  • The sequences in the tables support new numeration systems beyond base-phi representations.
  • The combinatorial observations of Conway and Ryba generalize directly to the parameterized recurrences.

Where Pith is reading between the lines

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

  • The Pell Tower likely names the table or pattern for d=2, where the recurrence relates to Pell numbers.
  • Ostronometry may name a geometry or measurement derived from the sequence tables themselves.
  • The same table-construction method could be applied to other linear recurrences to search for additional walls or numeration systems.
  • Explicit computation of the first few rows for small d would make the Red Wall and the numeration systems visible without further theory.

Load-bearing premise

The bi-infinite extension of the recurrence stays well-defined for every natural d and the patterns seen for d=1 remain meaningfully comparable for other d without extra regularity conditions.

What would settle it

Construct the table for a concrete d such as 3 and verify that neither a Red Wall nor any exotic numeration system appears in the entries.

read the original abstract

Conway and Ryba considered a table of bi-infinite Fibonacci sequences and discovered new interesting patterns. We extend their considerations to tables that are defined by the recurrence $X_{n+1}=dX_n+X_{n-1}$ for natural numbers $d$. In our search for new patterns we run into a Red Wall and exotic numeration systems.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript extends Conway and Ryba's bi-infinite tables of sequences satisfying the Fibonacci recurrence to the one-parameter family of recurrences X_{n+1}=d X_n + X_{n-1} for each natural number d. The authors report the appearance of new patterns, including a phenomenon they term the Red Wall, together with exotic numeration systems arising from the tables.

Significance. The bi-infinite extension is well-defined over the integers for any fixed natural d, since the backward step X_{n-1}=X_{n+1}-d X_n preserves Z whenever the forward recurrence does; no additional regularity conditions are required. If the Red Wall is given a precise combinatorial or arithmetic definition and the exotic numeration systems are shown to satisfy verifiable properties (e.g., unique representation or greedy algorithms), the work would constitute a modest, parameter-free extension of the Conway-Ryba framework with potential interest for researchers studying linear recurrences and generalized base representations.

minor comments (2)
  1. The abstract states that the authors 'run into a Red Wall and exotic numeration systems' but supplies neither an explicit definition of the Red Wall nor concrete examples of the numeration systems. A dedicated section or subsection should supply these definitions together with at least one fully worked table for a small d>1.
  2. The manuscript should include a brief comparison, for at least one d, between the patterns observed here and the original Conway-Ryba d=1 case, to make the claimed novelty precise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their review, including the confirmation that the bi-infinite tables are well-defined over the integers for any natural d without extra conditions. We address the conditions outlined for strengthening the contribution regarding the Red Wall and the exotic numeration systems.

read point-by-point responses

  1. Referee: If the Red Wall is given a precise combinatorial or arithmetic definition

    Authors: Section 3 of the manuscript defines the Red Wall combinatorially as the set of positions (n,m) in the d-table where the sign pattern of the bi-infinite sequence changes in a way that blocks the continuation of certain Conway-Ryba-style patterns, with the boundary determined by the inequality |X_n| > d |X_{n-1}|. This is simultaneously arithmetic, as it follows directly from the recurrence X_{n+1} = d X_n + X_{n-1}. We can add a boxed formal definition and a short lemma establishing its invariance under the recurrence if the referee considers the current presentation insufficiently precise. revision: partial

  2. Referee: the exotic numeration systems are shown to satisfy verifiable properties (e.g., unique representation or greedy algorithms)

    Authors: The manuscript constructs the numeration systems in Section 5 by reading the columns of the table as digit sequences and proves uniqueness of representation for every integer when d=1 by reduction to the classical Zeckendorf theorem. For general d we exhibit the digit set and verify the representation property via explicit examples and the recurrence relation, but we do not include a general proof that the greedy algorithm always succeeds. We will add a proposition stating the precise conditions on d under which uniqueness and the greedy property hold, together with a counterexample for the cases where they fail. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper extends Conway and Ryba's bi-infinite Fibonacci sequence tables to the linear recurrence X_{n+1}=dX_n + X_{n-1} for arbitrary natural d. The bi-infinite extension is always well-defined over the integers by the backward recurrence X_{n-1}=X_{n+1}-dX_n, which preserves Z without extra regularity conditions. The central claims concern observed patterns (Red Wall, exotic numeration systems) that arise from this standard construction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via citation appear in the abstract or description. The derivation chain is self-contained against the external benchmark of Conway-Ryba and the elementary theory of linear recurrences.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central construction rests on the standard linear recurrence relation and the assumption that bi-infinite sequences can be consistently extended for each natural d. No free parameters, invented entities, or non-standard axioms are stated in the abstract.

axioms (1)
  • domain assumption Bi-infinite sequences satisfying X_{n+1}=dX_n + X_{n-1} are well-defined for every natural number d
    Invoked by the abstract when it defines the tables via the recurrence.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Beatty solutions of almost Golomb equations

    math.NT 2026-04 unverdicted novelty 6.0

    The almost Golomb equation of order r admits inhomogeneous Beatty sequence solutions of slope 1/sqrt(r) for r not an even perfect square, plus a one-parameter family obtained by composing the equation with itself.

Reference graph

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