Half-Fibonacci sequence

Conjecture (1): The natural density of even terms in the sequence is 1/2.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-120424,
  title        = {Half-Fibonacci sequence},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-120424}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Conjecture (1): The natural density of even terms in the sequence is $1/2$ .

conjecture2. Conjecture (2): There are infinitely many consecutive pairs that differ by 1.

$a (0) = 1, a (1) = 3$ ; for $n \geq 2$ , $a (n) = f (a (n - 1)) + f (a (n - 2))$ where $f (x) = x /2$ if $x$ is even and $f (x) = x$ if $x$ is odd.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«120424» (2 statements).

theorem conjecture1 :
    Filter.Tendsto (fun n : ℕ => ((Finset.filter (fun k => a k % 2 = 0) (Finset.range n)).card
        : ℝ) / (n : ℝ))
      Filter.atTop (nhds (1 / 2 : ℝ))

theorem conjecture2 :
    Set.Infinite {n : ℕ | a (n + 1) = a n + 1 ∨ a n = a (n + 1) + 1}

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A120424

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.