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A001285 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 1's and 2's.
(Formerly M0193 N0071)
51
1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Or, follow a(0), .., a(2^k-1) by its complement.

Equals limiting row of A161175. - Gary W. Adamson, Jun 05 2009

Parse A010060 into consecutive pairs: (01, 10, 10, 01, 10, 01,...); then apply the rules: (01 -> 1; 10 ->2), obtaining (1, 2, 2, 1, 2, 1, 1,...). - Gary W. Adamson, Oct 25 2010

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.

F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).

Dubickas, Artūras. On a sequence related to that of Thue-Morse and its applications. Discrete Math. 307 (2007), no. 9-10, 1082--1093.MR2292537 (2008b:11086)

G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105.

G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.

A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1023

J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA '98, Springer-Verlag, 1999, pp. 1-16.

Francoise Dejean, Sur un Theoreme de Thue, J. Combinatorial Theory, vol. 13 A, iss. 1 (1972) 90-99.

Michael Gilleland, Some Self-Similar Integer Sequences

M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.

N. J. A. Sloane, The first 1000 terms as a string

S. Wolfram, Source for short Thue-Morse generating code

Index entries for "core" sequences

FORMULA

a(2n)=a(n), a(2n+1)=3-a(n), a(0)=1. Also, a(k+2^m)=3-a(k) if 0<=k<2^m.

a(n) = 2-A010059(n) = 1/2*(3-(-1)^A000120(n)). - Ralf Stephan, Jun 20 2003

a(n)=sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=A001316(n) {mod 3}. - Benoit Cloitre, May 09 2004

MAPLE

A001285 := proc(n) option remember; if n=0 then 1 elif n mod 2 = 0 then A001285(n/2) else 3-A001285((n-1)/2); fi; end;

s := proc(k) local i, ans; ans := [ 1, 2 ]; for i from 0 to k do ans := [ op(ans), op(map(n->if n=1 then 2 else 1 fi, ans)) ] od; RETURN(ans); end; t1 := s(6); A001285 := n->t1[n]; # s(k) gives first 2^(k+2) terms

MATHEMATICA

Nest[ Flatten@ Join[#, # /. {1 -> 2, 2 -> 1}] &, {1}, 7] (* Robert G. Wilson v, Feb 26 2005 *)

PROG

(PARI) a(n)=1+subst(Pol(binary(n)), x, 1)%2

(PARI) a(n)=sum(k=0, n, binomial(n, k)%2)%3

(PARI) a(n)=hammingweight(n)%2+1 \\ Charles R Greathouse IV, Mar 26 2013

(Haskell)

a001285 n = a001285_list !! n

a001285_list = map (+ 1) a010060_list

-- Reinhard Zumkeller, Oct 03 2012

CROSSREFS

Cf. A010060 for 0, 1 version, which is really the main entry for this sequence; also A003159. A001285(n)=1+A010060(n), A225186 (squares).

A026465 gives run lengths.

Cf. A010059 (1, 0 version).

Cf. A161175. - Gary W. Adamson, Jun 05 2009

Cf. A026430 (partial sums).

Boustrophedon transforms: A230958, A029885.

Sequence in context: A088569 A246144 A192763 * A088424 A097456 A164002

Adjacent sequences:  A001282 A001283 A001284 * A001286 A001287 A001288

KEYWORD

nonn,easy,core,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 1 20:00 EDT 2014. Contains 246317 sequences.