53175.lean`

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

9 declarations, 1 contain sorry, 0 declared as axioms

Catalan-Larcombe-French sequence

The Catalan-Larcombe-French sequence defined by $a(0)=1$, $a(1)=8$, and $$n^2 a(n) = 8(3n^2 - 3n + 1) a(n-1) - 128(n-1)^2 a(n-2)$$ for $n \ge 2$.

References:

Imports (1)

FormalConjecturesUtil

def OeisA53175.a Catalan-Larcombe-French sequence line 36 source
Source documentation
Catalan-Larcombe-French sequence.

standard axioms: propext what this means
Statement uses: Nat
2 instances, coercions and auxiliary constants left out.
```
def a : ℕ → ℕ
  | 0 => 1
  | 1 => 8
  | n + 2 =>
    let n' := n + 2
…
```
theorem OeisA53175.a_0 line 48 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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 OeisA53175.a_1 line 52 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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 OeisA53175.a_2 line 56 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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 OeisA53175.a_3 line 60 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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 OeisA53175.a_4 line 64 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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 OeisA53175.a_5 line 68 source
standard axioms: propext what this means
Statement uses: Eq, Nat, OeisA53175.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.
def OeisA53175.hankelMatrix The $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry $a(i+j)$ line 72 source
Source documentation
The $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry $a(i+j)$.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Matrix, Fin, HAdd.hAdd, OfNat.ofNat, Int
Proof uses: Equiv, Matrix.of, OeisA53175.a
12 instances, coercions and auxiliary constants left out.
```
def hankelMatrix (n : ℕ) : Matrix (Fin (n + 1)) (Fin (n + 1)) ℤ :=
…
```
theorem OeisA53175.conjecture Conjecture: let $P(n)$ be the $(n+1) \times (n+1)$ Hankel-type determinant with $(i,j)$-entry equal to $a(i+j)$ for all $i,j = 0, \ldots, n$ sorry line 84 source
Source documentation
Conjecture: let $P(n)$ be the $(n+1) \times (n+1)$ Hankel-type determinant with $(i,j)$-entry equal to $a(i+j)$ for all $i,j = 0, \ldots, n$. Then $P(n)/2^{n(n+3)}$ is a positive odd integer.
- Zhi-Wei Sun, Aug 14 2013
Proved by [ZS18], Theorem 1.2: $2^{-n(n+3)} |P_{i+j}|_{0 \le i,j \le n}$ is a positive odd integer for every $n \in \mathbb{N}$, where $P_n$ are the Catalan-Larcombe-French numbers.
depends on sorry what this means
Statement uses: Nat, Int, Matrix.det, Fin, HAdd.hAdd, OfNat.ofNat, OeisA53175.hankelMatrix, HPow.hPow, HMul.hMul, And, Dvd.dvd, LT.lt, HDiv.hDiv, Eq, HMod.hMod
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 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) :
    let detP := (hankelMatrix n).det
…
```
Mentions: OeisA53175.hankelMatrix