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import FormalConjecturesUtil
import FormalConjectures.OEIS.«105033»Conjectures associated with A102371
The sequence $a(n)$ is defined by the recurrence relation $a(1)=1$, and for $n>1$, $a(n) = a(n-1) \operatorname{XOR} (a(n-1) + n)$. The conjecture asks if $a(n) = 2^n - 1 - \operatorname{A105033}(n-1)$ for $n \ge 1$.
References:
namespace OeisA102371open Natopen Int
The primary defining sequence a.
$a(n)$ is defined by the recurrence relation $a(1)=1$, and for $n>1$,
$a(n) = a(n-1) \operatorname{XOR} (a(n-1) + n)$,
where $\operatorname{XOR}$ is the bitwise exclusive-or operator (^^^).
def a : ℕ → ℕ
| 0 => 0
| 1 => 1
| (n' + 2) =>
let anMinus1 := a (n' + 1)
anMinus1 ^^^ (anMinus1 + (n' + 2))Term theorems verifying the first few values of the sequence against the official OEIS b-file
@[category test, AMS 11]
theorem a_1 : a 1 = 1 := ⊢ a 1 = 1 All goals completed! 🐙@[category test, AMS 11]
theorem a_2 : a 2 = 2 := ⊢ a 2 = 2 All goals completed! 🐙@[category test, AMS 11]
theorem a_3 : a 3 = 7 := ⊢ a 3 = 7 All goals completed! 🐙@[category test, AMS 11]
theorem a_4 : a 4 = 12 := ⊢ a 4 = 12 All goals completed! 🐙@[category test, AMS 11]
theorem a_5 : a 5 = 29 := ⊢ a 5 = 29 All goals completed! 🐙Do we have $a(n) = 2^n - 1 - \operatorname{A105033}(n-1)$ for $n \ge 1$?
Note: The OEIS comment suggests $n-1$ which means this applies at least for $n \ge 1$, as we assume $\operatorname{A105033}(\mathbb{N})$ is defined on $\mathbb{N}$. We include the case $n=1$ which relies on $A105033(0)$, which is 0.
@[category research solved, AMS 11,
formal_proof using lean4 at
"https://github.com/KitaKen1/oeis-a102371/blob/d80fba9/lean/OeisA102371FC.lean#L490-L497"]
theorem conjecture : answer(True) ↔ ∀ n : ℕ, 0 < n → a n = 2^n - 1 - OeisA105033.a (n - 1) := ⊢ True ↔ ∀ (n : ℕ), 0 < n → OeisA102371.a n = 2 ^ n - 1 - OeisA105033.a (n - 1)
All goals completed! 🐙end OeisA102371