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
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 Apparently unpublished.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(-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
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof and Peter Boros, Jun 21 2004
STATUS
approved