A076826 a(n) = 2*(Sum_{k=0..n} A010060(k)) - n, where A010060 is a Thue-Morse sequence.

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

Offset: 0

Author

Benoit Cloitre, Nov 24 2002

Keywords

• nonn

Comments

Is there any interesting sequence b(n) such that b(n) mod 3 = a(n)?

Fixed point of the morphism 0->012; 1->1; 2->210 starting with a(0) = 0. - Philippe Deléham, Mar 14 2004

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8191
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 52.
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A000069 (odious numbers), A001969 (evil numbers).

Cf. A000120, A007413, A010060, A026430, A115384, A159481.

Programs

Mathematica

Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 1, 2}, 1 -> {1}, 2 -> {2, 1, 0}}) ]}], {0}, 6] (* Robert G. Wilson v, Mar 03 2005 *)
cnt=0; Join[{0}, Table[If[EvenQ[Count[IntegerDigits[n,2],1]], cnt--, cnt++ ]; cnt, {n,150}]] (* T. D. Noe, Jun 14 2007 *)

PARI

a(n)=if(n<0,0,2*sum(k=1,n,subst(Pol(binary(k)),x,1)%2)-n)

PARI

a(n)=if(n<1,0,if(n%2,1,if(n/2%2,2-a(n\4*2),a(n/2))))

Python

def A076826(n): return 1 if n&1 else (n.bit_count()&1)<<1 # Chai Wah Wu, Mar 01 2023

Formula

a(2k+1) = 1, a(4k) = a(2k), a(4k+2) = 2-a(2k). - Michael Somos, Dec 04 2002

a(2n) = 2*A010060(n); a(2n+1) = 1. - Benoit Cloitre, Mar 08 2004

a(n) = 2*(A026430(n+1) - 1) mod 3. - Philippe Deléham, Mar 28 2004

a(n) = (number of odious numbers <= n) - (number of evil numbers <= n) for n>0. - T. D. Noe, Jun 14 2007

a(n) = 2*A115384(n) - n. - Vladimir Shevelev, May 31 2009

a(n) = 0 if n and A000120(n) are even; a(n) = 2 if n is even but A000120(n) is odd; a(n) = 1 if n is odd. - Vladimir Shevelev, May 31 2009