176477.lean`

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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

5 declarations, 2 contain sorry, 0 declared as axioms

Recurrence with fourth powers of binomial coefficients

The sequence is defined by $a(1) = 2$, and for $n \ge 2$, $$(2n+1)^3 a(n) = 32n^3 a(n-1) + (21n^3 + 22n^2 + 8n + 1) \binom{2n-1}{n}^4.$$

References:

A176477
Z.-W. Sun, "Open Conjectures on Congruences", arXiv preprint arXiv:0911.5665 [math.NT], 2009-2011.
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

Imports (1)

FormalConjecturesUtil

def OeisA176477.a Rational recurrence sequence $a(n)$ line 39 source
Source documentation
Rational recurrence sequence $a(n)$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Rat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℚ
  | 0 => 0
  | 1 => 2
  | n + 2 =>
    let idx : ℚ := n + 2
…
```
theorem OeisA176477.a_1 Value of the sequence a at 1 line 52 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Rat, OeisA176477.a, OfNat.ofNat, Nat
Proof uses: 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_1 : a 1 = 2 := by rfl
…
```
Mentions: OeisA176477.a
theorem OeisA176477.a_2 Value of the sequence a at 2 line 56 source
Source documentation
Value of the sequence a at 2.

extra axioms: _private.0.OeisA176477.a_2._native.decide.ax_1 what this means
Statement uses: Eq, Rat, OeisA176477.a, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem a_2 : a 2 = 181 := by decide +native
…
```
Mentions: OeisA176477.a
theorem OeisA176477.conjecture1 sorry line 67 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, And, Eq, Rat.den, OeisA176477.a, LT.lt, Rat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
8 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem OeisA176477.conjecture2 sorry line 79 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Iff, And, Eq, Rat.den, OeisA176477.a, Odd, Int, Rat.num, Exists, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

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