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import FormalConjecturesUtilCatalan-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:
namespace OeisA53175Catalan-Larcombe-French sequence.
def a : ℕ → ℕ
| 0 => 1
| 1 => 8
| n + 2 =>
let n' := n + 2
let an1 := a (n + 1)
let an2 := a n
let term1 := 8 * (3 * n' ^ 2 - 3 * n' + 1) * an1
let term2 := 128 * (n' - 1) ^ 2 * an2
(term1 - term2) / (n' ^ 2)@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1
All goals completed! 🐙@[category test, AMS 11]
theorem a_1 : a 1 = 8 := ⊢ a 1 = 8
All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 80 := ⊢ a 2 = 80
All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 896 := ⊢ a 3 = 896
All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 10816 := ⊢ a 4 = 10816
All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 137728 := ⊢ a 5 = 137728
All goals completed! 🐙The $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry $a(i+j)$.
def hankelMatrix (n : ℕ) : Matrix (Fin (n + 1)) (Fin (n + 1)) ℤ :=
Matrix.of fun i j : Fin (n + 1) ↦ (a (i.val + j.val) : ℤ)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
@[category research open, AMS 11 15]
theorem conjecture (n : ℕ) :
let detP := (hankelMatrix n).det
let pow2 := (2 : ℤ) ^ (n * (n + 3))
pow2 ∣ detP ∧ 0 < detP / pow2 ∧ (detP / pow2) % 2 = 1 := n:ℕ⊢ let detP := (hankelMatrix n).det;
let pow2 := 2 ^ (n * (n + 3));
pow2 ∣ detP ∧ 0 < detP / pow2 ∧ detP / pow2 % 2 = 1
All goals completed! 🐙end OeisA53175