A191255 Fixed point of the morphism 0 -> 01, 1 -> 02, 2 -> 03, 3 -> 01.

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, 2

Offset: 1

Author

Clark Kimberling, May 28 2011

Keywords

• nonn

Comments

See A191250.

The asymptotic density of the occurrences of k = 0, 1, 2 and 3 is 1/2, 2/7, 1/7 and 1/14, respectively. The asymptotic mean of this sequence is 11/14. - Amiram Eldar, May 31 2024

Links

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

Crossrefs

Cf. A191250, A191254, A373157.

Positions of 0 or 3: A191257; positions of 0: A005408; positions of 1: A067368; positions of 2: A213258.

Programs

Mathematica

t = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0, 2}, 2 -> {0, 3}, 3 -> {0, 1}}] &, {0}, 9] (* this sequence *)
Flatten[Position[t, 0]] (* A005408, the odds *)
a = Flatten[Position[t, 1]] (* A067368 *)
b = Flatten[Position[t, 2]] (* A213258 *)
a/2  (* A191257 *)
b/4  (* a/2 *)

PARI

A191255(n) = if(n%2, 0, my(e=valuation(n, 2)%3); if(!e, 3, e)); \\ Antti Karttunen, May 28 2024, after Jianing Song's Sep 21 2018 formula

Formula

a(n) = 0 for odd n, otherwise a(n) is the unique number in {1,2,3} that is congruent to v2(n) modulo 3, where v2(n) = A007814(n) is the 2-adic valuation of n. - Jianing Song, Sep 21 2018 [Clarified by Jianing Song, May 30 2024]

Recurrence: a(2n-1) = 0, a(2n) = 1, 2, 3, 1 for a(n) = 0, 1, 2, 3 respectively. - Jianing Song, May 30 2024