102371.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

7 declarations, 1 contain sorry, 0 declared as axioms

Conjectures associated with A102371

The sequence $a(n)$ is defined by the recurrence relation $a(1)=1$, and for $n>1$, $a(n) = a(n-1) \operatorname{XOR} (a(n-1) + n)$. The conjecture asks if $a(n) = 2^n - 1 - \operatorname{A105033}(n-1)$ for $n \ge 1$.

References:

A102371

Imports (2)

def OeisA102371.a The primary defining sequence a line 43 source
Source documentation
The primary defining sequence a. $a(n)$ is defined by the recurrence relation $a(1)=1$, and for $n>1$, $a(n) = a(n-1) \operatorname{XOR} (a(n-1) + n)$, where $\operatorname{XOR}$ is the bitwise exclusive-or operator (^^^).

standard axioms: propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
| 0 => 0
| 1 => 1
| (n' + 2) =>
  let anMinus1 := a (n' + 1)
…
```
theorem OeisA102371.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file line 52 source
Source documentation
Term theorems verifying the first few values of the sequence against the official OEIS b-file

standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA102371.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_1 : a 1 = 1 := by rfl
…
```
Mentions: OeisA102371.a
theorem OeisA102371.a_2 line 55 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA102371.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102371.a_3 line 58 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA102371.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102371.a_4 line 61 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA102371.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102371.a_5 line 64 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA102371.a, OfNat.ofNat
Proof uses: Eq.refl
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA102371.conjecture sorry line 76 source
depends on sorry what this means
Statement uses: Iff, True, Nat, LT.lt, OfNat.ofNat, Eq, OeisA102371.a, HSub.hSub, HPow.hPow, OeisA105033.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

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