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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)
65
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.
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.
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
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1023 from T. D. Noe)
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.
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).
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
J. D. Currie, The Least Self-Shuffle of the Thue-Morse Sequence, J. Int. Seq. 17 (2014) # 14.10.2.
Françoise Dejean, Sur un Théorème de Thue, J. Combinatorial Theory, vol. 13 A, iss. 1 (1972) 90-99.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Arturas Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Volume 117, Issue 1, March 2006, Pages 222-239.
Arturas Dubickas, 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. 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.
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.
M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282)
N. J. A. Sloane, P. Flor, L. F. Meyers, G. A. Hedlund. M. Gardner, Collection of documents and notes related to A1285, A3270, A3324
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) = 1 + A010060(n).
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
G.f.: (3/(1 - x) - Product_{k>=0} (1 - x^(2^k)))/2. - Ilya Gutkovskiy, Apr 03 2019
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 *)
a[n_] := Mod[Sum[Mod[Binomial[n, k], 2], {k, 0, n}], 3]; Table[a[n], {n, 0, 101}] (* Jean-François Alcover, Jul 02 2019 *)
ThueMorse[Range[0, 120]]+1 (* Harvey P. Dale, May 07 2021 *)
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
(Python)
from itertools import islice
def A001285_gen(): # generator of terms
yield 1
blist = [1]
while True:
c = [3-d for d in blist]
blist += c
yield from c
A001285_list = list(islice(A001285_gen(), 30)) # Chai Wah Wu, Nov 13 2022
(Python)
def A001285(n): return 2 if n.bit_count()&1 else 1 # Chai Wah Wu, Mar 01 2023
CROSSREFS
Cf. A010060 for 0, 1 version, which is really the main entry for this sequence; also A003159. 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 A317960 A270642
KEYWORD
nonn,easy,core,nice
AUTHOR
STATUS
approved

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Last modified March 19 06:25 EDT 2024. Contains 370953 sequences. (Running on oeis4.)