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import FormalConjecturesUtilAlternating sum of signs of powers of $3/2$
The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$.
References:
namespace OeisA71532open Finset Filteropen scoped AsymptoticsThe sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$.
def a (n : ℕ) : ℤ :=
- ∑ k ∈ Icc 1 n, (-1 : ℤ) ^ (3 ^ k / 2 ^ k)
Value of the sequence a at 1.
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1
All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 0 := ⊢ a 2 = 0
All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 1 := ⊢ a 3 = 1
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 2 := ⊢ a 4 = 2
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 3 := ⊢ a 5 = 3
All goals completed! 🐙Is $a(n) > 0$ for all $n > 2$?
@[category research open, AMS 11]
theorem conjecture1 (n : ℕ) (hn : 2 < n) : 0 < a n := n:ℕhn:2 < n⊢ 0 < a n
All goals completed! 🐙For $n$ large enough, does $a(n) > \sqrt{n}$ always hold?
@[category research open, AMS 11]
theorem conjecture2 :
∃ N : ℕ, ∀ n : ℕ, N ≤ n → (a n : ℝ) > Real.sqrt (n : ℝ) := ⊢ ∃ N, ∀ (n : ℕ), N ≤ n → ↑(OeisA71532.a n) > √↑n
All goals completed! 🐙Conjecture: asymptotically, $a(n) \sim C \log(n)^2$ for some constant $C > 0$.
@[category research open, AMS 11]
theorem conjecture3 :
∃ C : ℝ, 0 < C ∧ (fun n : ℕ ↦ (a n : ℝ)) ~[atTop] (fun n : ℕ ↦ C * Real.log (n : ℝ) ^ 2) := ⊢ ∃ C, 0 < C ∧ (fun n ↦ ↑(a n)) ~[atTop] fun n ↦ C * Real.log ↑n ^ 2
All goals completed! 🐙Conjecture: the constant $C$ in $a(n) \sim C \log(n)^2$ is approximately $1.4$.
@[category research open, AMS 11]
theorem conjecture3_value :
let C : ℝ := answer(sorry)
|C - 1.4| < 0.1 ∧
(fun n : ℕ ↦ (a n : ℝ)) ~[atTop] (fun n : ℕ ↦ C * Real.log (n : ℝ) ^ 2) := ⊢ let C := sorry;
|C - 1.4| < 0.1 ∧ (fun n ↦ ↑(a n)) ~[atTop] fun n ↦ C * Real.log ↑n ^ 2
All goals completed! 🐙end OeisA71532