A092412 Fixed point of the morphism 0->11, 1->12, 2->13, 3->10, starting from a(1) = 1.

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

Offset: 1

Author

Philippe Deléham, Mar 22 2004

Keywords

• easy
• mult
• nonn

Links

• Paolo Xausa, Table of n, a(n) for n = 1..16384 (terms 1..1024 from Andrew Howroyd)
• Index entries for sequences that are fixed points of mappings.

Crossrefs

Cf. A001511, A033485, A035263, A346070 (same with values 3,0,1,2).

Programs

Mathematica

Nest[ Function[ l, {Flatten[(l /. {0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}})] }], {0}, 7] (* Robert G. Wilson v, Mar 04 2005 *)
SubstitutionSystem[{0 -> {1, 1}, 1 -> {1, 2}, 2 -> {1, 3}, 3 -> {1, 0}}, {1}, 7] // Last (* Jean-François Alcover, Sep 20 2019 *)
Mod[IntegerExponent[Range[100], 2] + 1, 4] (* Paolo Xausa, Feb 25 2025 *)

PARI

a(n)=(1 + valuation(n, 2)) %4; \\ Andrew Howroyd, Aug 06 2018

Python

def A092412(n): return (n&-n).bit_length()&3 # Chai Wah Wu, Jul 13 2022

Formula

a(n) = A001511(n) mod 4.

a(2n+1) = 1; a(2n) = a(n) + 1 mod 4.

a(n) == A035263(n) (mod 2); a(n) == A033485(n) (mod 2).

Multiplicative with a(2^e) = (1 + e) mod 4, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 06 2018

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 22/15. - Amiram Eldar, Nov 29 2022

Dirichlet g.f.: zeta(s)*(3*2^s+2^(2*s+1)+2^(3*s))/(1+2^s+4^s+8^s). - Amiram Eldar, Jan 04 2023