A095190 Doubled Thue-Morse sequence: a(2n) = A010060(n), a(2n+1) = A010060(n).

0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1

Offset

0,1

Comments

The A010060 sequence replacing 0 with 0,0 and 1 with 1,1.

Let n = Sum(c(k)*2^k), c(k) = 0,1, be the binary form of n, n = Sum(d(k)*3^k), d(k) = 0,1,2, the ternary form, n = Sum(e(k)*5^k), e(k) = 0,1,2,3,4, the base 5 form. Then a(n) = Sum(c(k)+d(k)) mod 2 = Sum(c(k)+e(k)) mod 2.

Links

Table of n, a(n) for n=0..104.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - N. J. A. Sloane, Sep 09 2018. See page 2 for a different construction of this same sequence.

F. Mignosi, A. Restivo, and M. Sciortino, Words and forbidden factors, WORDS (Rouen, 1999). Theoret. Comput. Sci. 273 (2002), no. 1-2, 99--117. MR1872445 (2002m:68096). [N. J. A. Sloane, Jul 10 2012]

Formula

a(n) = A096273(n) mod 2. - Benoit Cloitre, Jun 29 2004

a(n) = mod(A000120(floor(n/2)), 2) = mod(A010060(floor(n/2)), 2). - Paul Barry, Jan 07 2005

a(n) = mod(-1 + Sum_{k=0..n} mod(C(n, 2k), 2), 3). - Paul Barry, Jan 14 2005

a(n) = mod(log_2(Sum_{k=0..n} mod(C(n, 2k),2)),2). - Paul Barry, Jun 12 2006

Example

The Thue-Morse sequence is: 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ... so a(n) = 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 ...

Mathematica

a[n_] := Mod[DigitCount[Floor[n/2], 2, 1], 2]; Array[a, 100, 0] (* Amiram Eldar, Jul 28 2023 *)

Prog

(PARI) a(n)=hammingweight(n\2)%2 \\ Charles R Greathouse IV, May 08 2016

Crossrefs

Cf. A000120, A010059, A010060, A096273, A096288, A096289.

Sequence in context: A234045 A238468 A111113 * A131735 A131736 A152228

Adjacent sequences: A095187 A095188 A095189 * A095191 A095192 A095193

Keyword

easy,nonn

Author

Miklos Kristof and Peter Boros, Jun 21 2004

Status

approved