

A276397


Trajectory of 0 under the morphism 0 > 001, 1 > 0010.


3



0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0


COMMENTS

Or, fixed point of the morphism 0 > (0,0,1), 1 > (0,0,1,0).
Each 0 is replaced by the subsequence (0,0,1) and each 1 is replaced by the subsequence (0,0,1,0). It is easily seen that the only possible fixed point must start with 0. From there on the (initial segment of arbitrary length of the) fixed point can be obtained by simply iterating the map starting from this initial value.  M. F. Hasler, Oct 03 2016
The Beatty sequence for beta := (3 + sqrt(13))/2, A080081, has the property b(n+1)=b(n)+4 if n is already in the sequence, b(n+1) = b(n) + 3 otherwise. Here, every occurrence of "1" leads to an insertion of one more "0" (3 zeros instead of 2 zeros after the "1"). Therefore A080081(n)1 yields the index of the nth "1" in this sequence, i.e., A0800811 is the characteristic sequence of the present sequence.  M. F. Hasler, Oct 07 2016
Homogeneous Sturmian sequence with slope alpha = (sqrt(13)  3)/2 = 1/beta.  Michel Dekking, Feb 15 2019


LINKS

M. F. Hasler, Table of n, a(n) for n = 0..12969
J.P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 284.
T. C. Brown, A characterization of the quadratic irrationals, Canad. Math. Bull, 1991, 34(1), 3641.
Index entries for sequences that are fixed points of mappings


FORMULA

a(n) = floor((n+2)*alpha)  floor((n+1)*alpha), where alpha = (sqrt(13)3)/2.  Michel Dekking, Feb 15 2019


MATHEMATICA

Nest[ Flatten[ # /. {0 > {0, 0, 1}, 1 > {0, 0, 1, 0}}] &, {1}, 6]


PROG

(PARI) a=[0, 0, 1, 0]; while(#a<10^4, a=concat(t=apply(i>a[1..i+3], a))) \\ M. F. Hasler, Oct 03 2016


CROSSREFS

Different from A125117 and A144597.
Cf. A085550 ((sqrt(13)3)/2).
Sequence in context: A289001 A171588 A289035 * A286747 A131531 A022003
Adjacent sequences: A276394 A276395 A276396 * A276398 A276399 A276400


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Sep 11 2016


STATUS

approved



