2426.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

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

Central trinomial coefficients

Central trinomial coefficients: largest coefficient of $(1 + x + x^2)^n$, which is the coefficient of $x^n$ in the expansion of $(1 + x + x^2)^n$.

References:

A002426

Imports (1)

FormalConjecturesUtil

def OeisA2426.a Central trinomial coefficients: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \binom{2k}{k}$ line 35 source
Source documentation
Central trinomial coefficients: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \binom{2k}{k}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HAdd.hAdd, HDiv.hDiv, OfNat.ofNat, HMul.hMul, Nat.choose
8 instances, coercions and auxiliary constants left out.
```
def a (n : ℕ) : ℕ :=
…
```
theorem OeisA2426.a_0 Value of the sequence a at 0 line 40 source
Source documentation
Value of the sequence a at 0.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2426.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_0 : a 0 = 1 := by rfl
…
```
Mentions: OeisA2426.a
theorem OeisA2426.a_1 Value of the sequence a at 1 line 44 source
Source documentation
Value of the sequence a at 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2426.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: OeisA2426.a
theorem OeisA2426.a_2 Value of the sequence a at 2 line 48 source
Source documentation
Value of the sequence a at 2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2426.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_2 : a 2 = 3 := by rfl
…
```
Mentions: OeisA2426.a
theorem OeisA2426.a_3 Value of the sequence a at 3 line 52 source
Source documentation
Value of the sequence a at 3.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2426.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_3 : a 3 = 7 := by rfl
…
```
Mentions: OeisA2426.a
theorem OeisA2426.a_4 Value of the sequence a at 4 line 56 source
Source documentation
Value of the sequence a at 4.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Eq, Nat, OeisA2426.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_4 : a 4 = 19 := by rfl
…
```
Mentions: OeisA2426.a
theorem OeisA2426.conjecture An integer $n > 3$ is prime if and only if $a(n) \equiv 1 \pmod{n^2}$ sorry line 64 source
Source documentation
An integer $n > 3$ is prime if and only if $a(n) \equiv 1 \pmod{n^2}$. We have verified this for $n$ up to $8 \cdot 10^5$, and proved that $a(p) \equiv 1 \pmod{p^2}$ for any prime $p > 3$ (cf. A277640).
- Zhi-Wei Sun, Nov 30 2016
depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Iff, Nat.Prime, Nat.ModEq, HPow.hPow, OeisA2426.a
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem conjecture (n : ℕ) (hn : 3 < n) :
    n.Prime ↔ a n ≡ 1 [MOD n ^ 2] := by
…
```
Mentions: OeisA2426.a