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import FormalConjecturesUtilHalf-Fibonacci sequence
$a(0) = 1, a(1) = 3$; for $n \ge 2$, $a(n) = f(a(n-1)) + f(a(n-2))$ where $f(x) = x/2$ if $x$ is even and $f(x) = x$ if $x$ is odd.
References:
namespace OeisA120424Halve even numbers, leave odd numbers unchanged.
def f (x : ℕ) : ℕ := if x % 2 = 0 then x / 2 else xHalf-Fibonacci sequence starting with $1, 3$.
def a : ℕ → ℕ
| 0 => 1
| 1 => 3
| n + 2 => f (a (n + 1)) + f (a n)
Value of the sequence a at 0.
@[category test, AMS 11]
theorem a_0 : a 0 = 1 := ⊢ a 0 = 1 All goals completed! 🐙
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 3 := ⊢ a 1 = 3 All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 4 := ⊢ a 2 = 4 All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 5 := ⊢ a 3 = 5 All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 7 := ⊢ a 4 = 7 All goals completed! 🐙Conjecture (1): The natural density of even terms in the sequence is $1/2$.
@[category research open, AMS 11]
theorem conjecture1 :
Filter.Tendsto (fun n : ℕ => ((Finset.filter (fun k => a k % 2 = 0) (Finset.range n)).card
: ℝ) / (n : ℝ))
Filter.atTop (nhds (1 / 2 : ℝ)) := ⊢ Filter.Tendsto (fun n ↦ ↑{k ∈ Finset.range n | a k % 2 = 0}.card / ↑n) Filter.atTop (nhds (1 / 2))
All goals completed! 🐙Conjecture (2): There are infinitely many consecutive pairs that differ by 1.
@[category research open, AMS 11]
theorem conjecture2 :
Set.Infinite {n : ℕ | a (n + 1) = a n + 1 ∨ a n = a (n + 1) + 1} := ⊢ {n | a (n + 1) = a n + 1 ∨ a n = a (n + 1) + 1}.Infinite
All goals completed! 🐙end OeisA120424