A191254 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 01.

0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Author

Clark Kimberling, May 28 2011

Keywords

• nonn

Comments

For related sequences, see notes in the Mathematica program.

The asymptotic density of the occurrences of k = 0, 1 and 2 is 1/2, 1/3 and 1/6, respectively. The asymptotic mean of this sequence is 2/3. - Amiram Eldar, May 31 2024

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A191250, A191255.

Positions of 0 or 2: A003159; positions of 0: A005408; positions of 1: A036554; positions of 2: A108269.

Programs

Mathematica

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 1}}] &, {0}, 9]  (* A191254 *)
Flatten[Position[t, 0]]  (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* A036554 *)
b = Flatten[Position[t, 2]] (* A108269 *)
a/2 (* A003159 *)
b/4 (* A003159 *)

PARI

A191254(n) = if(n%2,0,if(valuation(n,2)%2,1,2)); \\ Antti Karttunen, Nov 06 2018

Formula

From Jianing Song, May 30 2024: (Start)

Recurrence: a(2n-1) = 0, a(2n) = 1, 2, 1 for a(n) = 0, 1, 2 respectively.

a(n) = 0 for odd n; a(n) = 1 for even n such that v2(n) is odd; a(n) = 2 for even n such that v2(n) is even, where v2(n) = A007814(n) is the 2-adic valuation of n. (End)