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A056832 All a(n) = 1 or 2; a(1) = 1; get next 2^k terms by repeating first 2^k terms and changing last element so sum of first 2^(k+1) terms is odd. 11
1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Dekking (2016) calls this the Toeplitz sequence or period-doubling sequence. - N. J. A. Sloane, Nov 08 2016

Fixed point of the morphism 1->12 and 2->11 (1 ->12 ->1211 ->12111212 ->..). - Benoit Cloitre, May 31 2004

a(n) is multiplicative. - Christian G. Bower, Jun 03 2005

a(n) is the least k such that A010060(n-1+k)=1-A010060(n-1); the sequence {a(n+1)-1} is the characteristic sequence for A079523. - Vladimir Shevelev, Jun 22 2009

REFERENCES

Karamanos, Kostas. "From symbolic dynamics to a digital approach." International Journal of Bifurcation and Chaos 11.06 (2001): 1683-1694. (Full version. See p. 1685)

Karamanos, K. (2000). From symbolic dynamics to a digital approach: chaos and transcendence. In Michel Planat (Ed.), Noise, Oscillators and Algebraic Randomness (Lecture Notes in Physics, pp. 357-371). Springer, Berlin, Heidelberg. (Short version. See p. 359)

M. R. Schroeder, Fractals, Chaos, Power Laws, W. H. Freeman, NY, 1991; pp. 277-279.

M. R. Schroeder, Letter to N. J. A. Sloane, May 05 1994.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

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

M. R. Schroeder, Letter to N. J. A. Sloane, May 05 1994.

Index entries for sequences that are fixed points of mappings

FORMULA

a(n) = ((-1)^(n+1)*A002425(n)) modulo 3. - Benoit Cloitre, Dec 30 2003

a(1)=1, a(n) = 1 + {sum(i=1, n-1, a(i)*a(n-i)) mod 2}. - Benoit Cloitre, Mar 16 2004

a(n) is multiplicative with a(2^e)=1+(1-(-1)^e)/2, a(p^e)=1 if p>2. - Michael Somos, Jun 18 2005

[a(2^n+1) .. a(2^(n+1)-1)] = [a(1) .. a(2^n-1)]; a(2^(n+1))=3-a(2^n).

For n>0, a(n) = 2-A035263(n). - Benoit Cloitre, Nov 24 2002

a(n)=2 if n-1 is in A079523; a(n)=1 otherwise. - Vladimir Shevelev, Jun 22 2009

a(n) = A096268(n-1) + 1. - Reinhard Zumkeller, Jul 29 2014

EXAMPLE

1 -> 1,2 -> 1,2,1,1 -> 1,2,1,1,1,2,1,2 -> 1,2,1,1,1,2,1,2,1,2,1,1,1,2,1,1.

Here we have 1 element, then 2 elements, then 4, 8, 16, etc.

MATHEMATICA

Nest[ Function[l, {Flatten[(l /. {1 -> {1, 2}, 2 -> {1, 1}})]}], {1}, 7] (* Robert G. Wilson v, Mar 03 2005 *)

Table[Mod[-(-1)^(n + 1) (-1)^n Numerator[EulerE[2 n + 1, 1]], 3] , {n, 0, 120}] (* Michael De Vlieger, Aug 15 2016, after Jean-Fran├žois Alcover at A002425 *)

PROG

(PARI) a(n)=numerator(2/n*(4^n-1)*bernfrac(2*n))%3

(PARI) a(n)=if(n<1, 0, valuation(n, 2)%2+1) /* Michael Somos, Jun 18 2005 */

(Haskell)

a056832 n = a056832_list !! (n-1)

a056832_list = 1 : f [1] where

   f xs = y : f (y : xs) where

          y = 1 + sum (zipWith (*) xs $ reverse xs) `mod` 2

-- Reinhard Zumkeller, Jul 29 2014

CROSSREFS

Cf. A197911 (partial sums).

Essentially same as first differences of Thue-Morse, A010060. - N. J. A. Sloane, Jul 02 2015

See A035263 for an equivalent version.

See also A002425, A079523, A096268.

Limit of A317956(n) for large n.

Sequence in context: A322028 A079806 A045887 * A105931 A279495 A300409

Adjacent sequences:  A056829 A056830 A056831 * A056833 A056834 A056835

KEYWORD

easy,nonn,nice,mult

AUTHOR

Jonas Wallgren, Aug 30 2000

STATUS

approved

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Last modified October 13 19:48 EDT 2019. Contains 327981 sequences. (Running on oeis4.)