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import FormalConjecturesUtilFibonacci transform satisfying $|a(n)| = F(n+1)$
The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$. The sequence takes values in $\mathbb{Z}$.
References:
arxiv/2605.22763 Advancing Mathematics Research with AI-Driven Formal Proof Search by George Tsoukalas et al.
namespace OeisA103311The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$. The sequence takes values in $\mathbb{Z}$.
def a : ℕ → ℤ
| 0 => 0
| 1 => 1
| 2 => 1
| 3 => 0
| n + 4 => 3 * a (n + 3) - 4 * a (n + 2) + 2 * a (n + 1) - a n@[category test, AMS 11]
lemma a_0 : a 0 = 0 := ⊢ a 0 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_2 : a 2 = 1 := ⊢ a 2 = 1 All goals completed! 🐙@[category test, AMS 11]
lemma a_3 : a 3 = 0 := ⊢ a 3 = 0 All goals completed! 🐙@[category test, AMS 11]
lemma a_4 : a 4 = -2 := ⊢ a 4 = -2 All goals completed! 🐙Fibonacci transform satisfying $|a(n)| = F(n+1)$.
Conjecture: all elements in absolute value are Fibonacci numbers.
A formal proof has been found with the methods described in arxiv/2605.22763.
@[category research solved, AMS 11, formal_proof using formal_conjectures at
"https://github.com/mo271/formal-conjectures/blob/a32396489dcb8f86c3549b93aa358ac6a10a3a1f/FormalConjectures/OEIS/103311.wip.lean#L218"]
theorem a_abs_eq_fib (n : ℕ) : ∃ m : ℕ, Int.natAbs (a n) = Nat.fib m := n:ℕ⊢ ∃ m, (a n).natAbs = Nat.fib m
All goals completed! 🐙end OeisA103311