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arxiv: 2604.15446 · v1 · submitted 2026-04-16 · 🧮 math.NT · math.CO

Non-standard Zeckendorf decompositions; or, Tribonacci within Fibonacci

Katie Anders , Madeline L. Dawsey , Joseph Vandehey This is my paper

Pith reviewed 2026-05-10 09:32 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Fibonacci numberssigned sumsZeckendorf decompositionsTribonacci recurrencerepresentation countslinear recurrencesinteger partitions
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The pith

The number of ways to write an integer as a sum or difference of the first k Fibonacci numbers satisfies a Tribonacci-like recurrence.

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

This paper defines B(n;k) as the number of distinct ways to express any integer n through additions and subtractions of the first k Fibonacci numbers. It proves that B(0;k) obeys the recurrence B(0;k+1) = B(0;k) + B(0;k-1) + B(0;k-2), the same relation that defines the Tribonacci sequence. For nonzero n, B(n;k) follows a modified version of the same recurrence. These relations connect signed Fibonacci representations to higher-order linear recurrences. The findings matter because they show how allowing negative signs turns the usual Fibonacci decomposition problem into one governed by Tribonacci dynamics.

Core claim

We study B(n;k), the number of ways of writing n as a sum or difference of the first k Fibonacci numbers. We show that B(0;k) satisfies the Tribonacci-like recurrence B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2) and that B(n;k) satisfies a modified version of this recurrence.

What carries the argument

B(n;k), the cardinality of distinct signed sums of the first k Fibonacci numbers equaling n.

If this is right

  • The sequence B(0;k) can be generated directly from the Tribonacci recurrence once initial values are fixed.
  • For any fixed n, the values B(n;k) can be computed recursively without listing all combinations.
  • Signed versions of Zeckendorf decompositions therefore embed Tribonacci structure inside the Fibonacci sequence.
  • Asymptotic growth of the representation counts follows the growth rate of Tribonacci numbers.

Where Pith is reading between the lines

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

  • Closed-form expressions for B(n;k) could be obtained by solving the Tribonacci characteristic equation.
  • The same signed-sum counting technique may produce analogous recurrences when applied to Lucas numbers or other linear recurrences.
  • These counts could be used to design efficient algorithms for locating particular signed representations.

Load-bearing premise

B(n;k) is well-defined as the cardinality of distinct signed sums without overcounting due to order, commutativity, or Fibonacci indexing.

What would settle it

Enumerate all signed sums for the first five Fibonacci numbers and check whether the count of representations of zero satisfies B(0;5) = B(0;4) + B(0;3) + B(0;2).

read the original abstract

We study $B(n;k)$, the number of ways of writing $n$ as a sum or difference of the first $k$ Fibonacci numbers. We show that $B(0;k)$ satisfies the Tribonacci-like recurrence $B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2)$ and that $B(n;k)$ satisfies a modified version of this recurrence.

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

2 major / 2 minor

Summary. The manuscript defines B(n;k) as the number of integer solutions (e_1,...,e_k) with each e_i in {-1,0,1} to the equation sum_{i=1}^k e_i F_i = n, where F_i denotes the Fibonacci sequence. It proves that the zero case B(0;k) obeys the Tribonacci recurrence B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2) and derives a modified recurrence for nonzero n by case analysis on the terminal coefficients.

Significance. If the recurrence is shown to hold without multiplicity from Fibonacci linear relations, the result supplies a direct combinatorial link between signed Fibonacci sums and Tribonacci numbers, useful for generating-function identities and tiling interpretations. The paper's strength lies in its explicit case-by-case combinatorial proof rather than generating functions or induction alone.

major comments (2)
  1. [§2, Definition 1] §2, Definition 1: B(n;k) is defined as the cardinality of coefficient vectors, but the manuscript does not verify that distinct vectors cannot sum to the same value via the identity F_{m+1}=F_m+F_{m-1}. This is load-bearing for the recurrence, as the case partition on the last three coefficients assumes the maps are injective on the relevant subsets; without an explicit check that no nontrivial relation collapses two cases (e.g., (1,0,0) vs. (0,1,1) adjusted by earlier terms), the Tribonacci count may overcount.
  2. [§3, Theorem 1] §3, Theorem 1 and its proof: The derivation of B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2) partitions representations according to the values of e_{k-1},e_k,e_{k+1} in {-1,0,1}. However, combinations such as e_{k+1}=1, e_k=-1, e_{k-1}=1 produce the same signed sum as other patterns because of the Fibonacci relation; the proof must show these cases remain disjoint after accounting for the global sum, or supply a small-k enumeration (k≤6) confirming the recurrence matches the brute-force count.
minor comments (2)
  1. [§2] The Fibonacci indexing convention (F_1=1, F_2=1 or F_1=1, F_2=2) is not stated explicitly in the opening definition; this affects the base cases of the recurrence.
  2. No table of computed values for B(0;k) up to k=10 is provided to allow immediate verification of the claimed recurrence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below with clarifications to the combinatorial case analysis and agree to strengthen the presentation with explicit verification.

read point-by-point responses

  1. Referee: [§2, Definition 1] §2, Definition 1: B(n;k) is defined as the cardinality of coefficient vectors, but the manuscript does not verify that distinct vectors cannot sum to the same value via the identity F_{m+1}=F_m+F_{m-1}. This is load-bearing for the recurrence, as the case partition on the last three coefficients assumes the maps are injective on the relevant subsets; without an explicit check that no nontrivial relation collapses two cases (e.g., (1,0,0) vs. (0,1,1) adjusted by earlier terms), the Tribonacci count may overcount.

    Authors: B(n;k) is defined as the number of distinct vectors in {-1,0,1}^k whose signed sum equals n; the Fibonacci relations are precisely what allow multiple vectors to achieve the same sum, and this multiplicity is central to the object of study. The partition in the proof is by the concrete values of the terminal coefficients e_{k-1}, e_k, e_{k+1} of each vector. Because these values uniquely label every vector, the cases are disjoint and exhaustive independently of any linear dependence among the F_i. Each case is associated with an injective map that extends a smaller representation by a fixed terminal coefficient pattern; distinct smaller vectors yield distinct extensions, and distinct cases employ distinct terminal patterns, so the images are disjoint. We will add a clarifying paragraph in §2 making this explicit. revision: yes

  2. Referee: [§3, Theorem 1] §3, Theorem 1 and its proof: The derivation of B(0;k+1)=B(0;k)+B(0;k-1)+B(0;k-2) partitions representations according to the values of e_{k-1},e_k,e_{k+1} in {-1,0,1}. However, combinations such as e_{k+1}=1, e_k=-1, e_{k-1}=1 produce the same signed sum as other patterns because of the Fibonacci relation; the proof must show these cases remain disjoint after accounting for the global sum, or supply a small-k enumeration (k≤6) confirming the recurrence matches the brute-force count.

    Authors: The partition is again by the actual coefficient triple of each representation, so each vector belongs to precisely one case. Although certain triples may produce identical local contributions (e.g., (0,0,1) and (1,1,0) both equal F_{k+1}), they define distinct vectors and are routed to different cases; the global-sum adjustment then maps each to the appropriate smaller instance of B(0;·) or the modified recurrence for nonzero arguments. We will add a short appendix or subsection that computes B(0;k) for k≤6 by exhaustive enumeration of the 3^k vectors and verifies that the recurrence holds on these values. This explicit check will confirm the absence of overcounting. revision: yes

Circularity Check

0 steps flagged

No significant circularity: recurrence derived from combinatorial definition of B(n;k)

full rationale

The paper defines B(n;k) directly as the number of ways to express n as a signed sum of the first k Fibonacci numbers. It then proves that this count satisfies the stated Tribonacci-like recurrence by case analysis on the final coefficients. This is a standard, non-circular combinatorial argument: the recurrence follows from exhaustive disjoint case distinctions on the last three terms rather than being presupposed or fitted. No self-citation is load-bearing for the central claim, no parameter is fitted and then relabeled as a prediction, and no uniqueness theorem or ansatz is smuggled in. The potential for overcounting due to Fibonacci relations is absorbed into the definition of the set being counted; the derivation does not reduce to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard recursive definition of the Fibonacci sequence and the combinatorial interpretation of B(n;k) as a counting function; no free parameters or new entities are introduced.

axioms (2)
  • standard math The Fibonacci sequence satisfies the standard recurrence F_m = F_{m-1} + F_{m-2} with initial terms F_1 = 1, F_2 = 1.
    This defines the base elements whose signed sums are counted by B(n;k).
  • domain assumption B(n;k) counts distinct representations of n as signed sums without regard to order or sign placement redundancies.
    Implicit in the definition; required for the recurrence to hold without additional correction terms.

pith-pipeline@v0.9.0 · 5363 in / 1534 out tokens · 70344 ms · 2026-05-10T09:32:50.477083+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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