105033.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

6 declarations, 0 contain sorry, 0 declared as axioms

Sloping binary numbers: read array of binary numbers (right-justified) along diagonals of slope $-1$

References:

A105033

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/OEIS/102371.lean

def OeisA105033.a The primary defining sequence a line 37 source
Source documentation
The primary defining sequence a. a n is defined by the formula: $$a(n) = n - \sum_{k \ge 0, 2^{k+1} \le n, n \equiv k \pmod{2^{k+1}}} 2^{k+1}$$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: HSub.hSub, Finset.sum, Finset.range, ite, And, LE.le, HPow.hPow, OfNat.ofNat, HAdd.hAdd, Eq, HMod.hMod
16 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA105033.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 42 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105033.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_0 : a 0 = 0 := by decide
…
```
Mentions: OeisA105033.a
theorem OeisA105033.a_1 line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105033.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105033.a_2 line 48 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105033.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105033.a_3 line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105033.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA105033.a_4 line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA105033.a, OfNat.ofNat
Proof uses: of_decide_eq_true, id, Bool, Decidable.decide, Bool.true, Eq.refl
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.