Metadata
bibkey: sobolewskiulas2022metafibonacci
authors: Bartosz Sobolewski and Maciej Ulas
year: 2022
title: Solutions of certain meta-Fibonacci recurrences
doi: 10.4064/aa220613-19-1
url: https://arxiv.org/abs/2204.04011v1
claim: "Question 7.5 asks whether the increasing occurrence enumerations of each parity of h_{1,1} are k-regular for some k."
strata_touched:
- D5/S1/Recurrence/MetaFibonacci/SobolewskiUlasParityPositions
license: citation-only
triage: anchor
Verified locator
DOI: 10.4064/aa220613-19-1
Source: https://arxiv.org/abs/2204.04011v1
The journal publication is Acta Arithmetica 209 (2023), 269–289. The quotations and page numbers below refer to arXiv v1. The journal text has not been verified; agreement with v1 and completeness of the prior-art screen are ASSUMED-UNVERIFIED. No priority claim is made.
Source definitions
Page 4: “We write h(n) = h_{1,1}(n) and in the sequel, we consider the sequence h = (h(n)){n∈ℕ}.” Page 3 defines h{a,b}; taking a=b=1 gives h(n)=1 for n≤1 and h(n)=h(n−h(n−1))+h(n−2) for n>1.
Page 8: “Let us write r(n) = h(n) (mod 2).”
Section 2, page 4: “Let k ≥ 2 be a fixed integer.” The k-kernel is “K_k(a) = {(a_{k^j n+i})_{n∈ℕ} : j ∈ ℕ, 0 ≤ i < k^j}”. “In other words, the ℤ-submodule generated by K_k(a) is finitely generated.”
The Lean sequence has natural arguments, with negative recursive arguments sent to zero by truncated subtraction. This preserves their value one. Its termination minimum is inactive because every sequence value is positive. Occurrence enumeration uses Nat.nth and starts at index zero.
Question 7.5
Page 16: “Let u_i = (u_i(n)) for i = 0, 1 be the increasing sequence of nonnegative integers such that r(u_i(n)) = i. Is the sequence u_i is k-regular for some k?” The repeated “is” is present in v1.
Page 17: “According to our numerical computations (using the mentioned package) both sequences u_0, u_1 are 2-regular.”
The formal result chooses base two for both parities. Its mechanism is exact eight-per-sixteen block balance and a finite-state correction to twice the enumeration index. Other parameters of the recurrence and other moduli are outside the theorem’s scope.