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import FormalConjecturesUtil
namespace OeisA103425
The primary defining sequence a.
$a(n)$ is defined by the recurrence relation $a(n) = 3 a(n-1) + a(n-2) - 3 a(n-3)$
with initial terms $a(0)=1, a(1)=3, a(2)=5$.
def a : ℕ → ℕ
| 0 => 1
| 1 => 3
| 2 => 5
| n + 3 => 3 * a (n + 2) + a (n + 1) - 3 * a nTerm theorems verifying the first few values of the sequence against the official OEIS b-file
@[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 = 3 := ⊢ a 1 = 3 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 5 := ⊢ a 2 = 5 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 15 := ⊢ a 3 = 15 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 41 := ⊢ a 4 = 41 All goals completed! 🐙def IsWeightedTribonacci (a b c : ℤ) (x : ℕ → ℤ) : Prop :=
∀ n, x (n + 3) = a * x (n + 2) + b * x (n + 1) + c * x nThe current sequence contains primes, including $3, 5, 41, 21523361$. Is there an $(a, b, c)$ weighted tribonacci sequence with $a, b, c$ relatively prime which is prime-free?
@[category research open, AMS 11]
theorem conjecture : answer(sorry) ↔
∃ (a b c : ℤ) (x : ℕ → ℤ),
Nat.gcd (Int.gcd a b) c.natAbs = 1 ∧
IsWeightedTribonacci a b c x ∧
∀ n, ¬ (x n).natAbs.Prime := ⊢ True ↔ ∃ a b c x, (a.gcd b).gcd c.natAbs = 1 ∧ IsWeightedTribonacci a b c x ∧ ∀ (n : ℕ), ¬Nat.Prime (x n).natAbs
All goals completed! 🐙end OeisA103425