bibkey: guo2025apwenian authors: Ying-Jun Guo, Guo-Niu Han year: 2025 title: “On a family of automatic apwenian sequences” doi: 10.1016/j.disc.2025.114399 url: https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf claim: “Families of 2-automatic apwenian sequences generated by directive sequences of morphisms, the classification of purely automatic apwenian sequences over {0,1,2} as the period-doubling-like sequences, and Conjectures 1 and 2 on general alphabets.” strata_touched:
- D5/S3/Combinatorics/Apwenian/GuoHan
- D5/S3/Combinatorics/Apwenian/GuoHanGeneration license: citation-only triage: anchor
Guo and Han, a family of automatic apwenian sequences
An integer sequence a is apwenian if a(0) = 1 and a(n) ≡ a(2n+1) + a(2n+2) (mod 2) for every n; equivalently, all its Hankel determinants are odd. The paper studies automatic apwenian sequences. Its Theorem 2 shows that a purely automatic sequence over {0, 1, 2}, that is a fixed point of a uniform morphism, is apwenian exactly when its parity sequence is the period-doubling sequence, and Conjecture 2 asserts the same for every finite alphabet whose only odd letter is 1.
The module D5/S3/Combinatorics/Apwenian/GuoHan proves Conjecture 2.
Conjecture 1: finite-alphabet level generation
Section 3, Conjecture 1, printed page 6 of the author manuscript states:
Let {u(n)}ₙ≥₀ be a 2-automatic apwenian sequence over {0,1}. Then there exist a finite alphabet Σ ⊂ N and a sequence σ over A(Σ) such that {u(n)}ₙ≥₀ = ϕ(σ) (mod 2).
Here A(Σ) consists of free-monoid endomorphisms taking every alphabet letter to a two-letter word over Σ whose letter sum has the original letter’s parity. Starting from X₀ = [1], the levels satisfy Xⱼ₊₁ = σⱼ(Xⱼ), and ϕ(σ) is the successive concatenation X₀X₁X₂⋯. This is a level construction; neither prolongability nor a fixed-point equation is part of the assertion. Several odd letters are allowed.
The result D5/S3/Combinatorics/Apwenian/GuoHanGeneration.result gives a constant directive as a
stronger witness. Its source automaticity premise is a finite DFAO evaluated on canonical binary
expansions in most-significant-first order. Either the input [0] or the empty input may represent zero.
The sequence is natural-valued with every value less than two; unused machine outputs need not be binary.
The construction adjoins a fresh zero-loop state and uses the finite set of occurring adjacent state pairs sₙ = (a(n), a(n+1)). The ordered children are L(q,r) = (δ₁(q), δ₀(r)) and R(q,r) = (δ₀(r), δ₁(r)); they send sₙ to s₂ₙ₊₁ and s₂ₙ₊₂. Enumerating this occurring set with its root at index zero gives injective labels e(s) = 2i(s) + ε(s), where ε is the first state’s output parity. The root label is one, and τ(e(s)) = [e(L(s)), e(R(s))] is sum-equivalent by the apwenian recurrence on the occurring pairs. The exact ordered j-th level is [e(s₂ʲ₋₁₊ₖ)] for 0 ≤ k < 2ʲ. Concatenating the first j levels gives exactly the first 2ʲ − 1 values of v(n) = e(sₙ). These lengths are unbounded, so this defines a unique generated sequence, and v(n) modulo two equals u(n) for every n.
The author and institutional manuscripts agree on the inspected definitions and conjecture. The publisher version-of-record body remains unverified. The literature screen recorded in preregistration issue 13031 found no equivalent settlement in its bounded author bibliography, HAL metadata, Crossref, Semantic Scholar, arXiv and repository scopes. This is a bounded finding and makes no worldwide-priority claim.
Verified locator
DOI: 10.1016/j.disc.2025.114399
URL: https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf
- Locator: Section 4, Theorem 2, Lemmas 4–8 and Conjecture 2 (author version).
- Locator: Section 3, definitions of A(Σ) and ϕ, and Conjecture 1, printed page 6 (author version).
- Institutional manuscript: https://cnrs.hal.science/hal-04923519/file/p120autoapw.pdf?download=1