A105931 a(1) = 1 then a(n) = a(n-1) - (-1)^ceiling(n/2)*a(floor(n/2)).

1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2

Offset: 1

Author

Benoit Cloitre, Apr 26 2005

Keywords

• sign
• mult

Comments

The asymptotic density of the occurrences of -1, 1, and 2 are 1/6, 1/2, and 1/3, respectively. - Amiram Eldar, Nov 30 2022

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A056832, A003159.

Programs

Mathematica

a[n_] := Module[{e = IntegerExponent[n, 2]}, If[e > 0, If[Mod[e, 2] == 1, 2, -1], 1]]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 after the second PARI code *)

PARI

a(n)=if(n<2,1,a(n-1)-(-1)^ceil(n/2)*a(floor(n/2)))

PARI

a(n)={my(e=valuation(n, 2)); if(e>0, if(e%2, 2, -1), 1)} \\ Andrew Howroyd, Aug 06 2018

Formula

abs(a(n)) = A056832(n).

a(k) = -1 iff k is in 4*A003159.

From Amiram Eldar, Nov 30 2022: (Start)

Multiplicative with a(2^e) = 2 if e is odd and -1 if e is even, and a(p^e) = 1 for p > 2.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)

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