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import FormalConjecturesUtilNumber of times $n$ occurs as a binary sub-pattern of $n^2$
The sequence $a(n)$ is the number of times the binary expansion of $n$ appears as a contiguous sublist (infix) in the binary expansion of $n^2$.
References:
namespace OeisA76141The binary representation of a natural number $n$, most significant bit first. For $n = 0$, this is $[0]$.
def binaryPattern (n : ℕ) : List ℕ :=
if n = 0 then [0] else (Nat.digits 2 n).reverseNumber of times the binary pattern of $n$ occurs as an infix of the binary pattern of $n^2$.
def a (n : ℕ) : ℕ :=
let pat := binaryPattern n
let tgt := binaryPattern (n ^ 2)
tgt.tails.countP (pat.isPrefixOf ·)
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 = 1 := ⊢ a 1 = 1
All goals completed! 🐙
Value of the sequence a at 2.
@[category test, AMS 11]
theorem a_2 : a 2 = 1 := ⊢ a 2 = 1
All goals completed! 🐙
Value of the sequence a at 3.
@[category test, AMS 11]
theorem a_3 : a 3 = 0 := ⊢ a 3 = 0
All goals completed! 🐙
Value of the sequence a at 4.
@[category test, AMS 11]
theorem a_4 : a 4 = 1 := ⊢ a 4 = 1
All goals completed! 🐙
Value of the sequence a at 5.
@[category test, AMS 11]
theorem a_5 : a 5 = 0 := ⊢ a 5 = 0
All goals completed! 🐙Is $a(n) \le 1$ for all $n$?
@[category research open, AMS 11]
theorem conjecture (n : ℕ) : a n ≤ 1 := n:ℕ⊢ a n ≤ 1
All goals completed! 🐙end OeisA76141