|
| |
|
|
A026465
|
|
Length of n-th run of identical symbols in A001285 (Thue-Morse sequence).
|
|
11
| |
|
|
1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,2
|
|
|
COMMENTS
| Number of representations of n as a sum of Jacobsthal numbers (1 is allowed twice as a part). Partial sums are A003159. With interpolated zeros, g.f. is Product{k>=1, 1+x^A078008(k)}/2. - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
Can also be generated by counting the consecutive 0's or 1's in A010060 or A010059. - Robin D. Saunders (saunders_robin_d(AT)hotmail.com), Sep 06 2006
The sequence (starting with the second term) can also be calculated by the following method, Carlo Carminati, Feb 25 2011: (Start)
Apply repeatedly to the string S_0=[2], the following algorithm: take a string S, double it, if the last figure is 1, just add the last figure to the previous one, if the last figure is greater than one, decrease it by one unit and concatenate a figure 1 at the end. (This algorithm is connected with the interpretation of the sequence as a continued fraction expansion) (End)
This sequence, starting with the second term, happens to be the continued fraction expansion of the biggest cluster point of the set {x in [0,1] : F^k(x) >= x, for all k in N}, where F denotes the Farey map - Carlo Carminati, Feb 28 2011
Starting with the second term, the fixed point of the substitution 2->211, 1->2. - Carlo Carminati, Mar 3 2011.
|
|
|
LINKS
| J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo, Dynamics of continued fractions and kneading sequences of unimodal maps, arXiv:1012.2131
S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math. 24 (1989), 83-96.
J. Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci. 218 (1999) 3-12.
Cor Kraaikamp, Thomas A. Schmidt, Wolfgang Steiner : Natural extensions and entropy of $\alpha$-continued fractions
|
|
|
FORMULA
| It appears that the sequence can be calculated by any of the following methods: (1) Start with 1 and repeatedly replace 1 with 1, 2, 1 and 2 with 1, 2, 2, 2, 1; (2) a(1)=1, all terms are either 1 or 2 and, for n>0, a(n)=1 if the length of the n-th run of 2's is 1; a(n)=2 if the length of the n-th run of consecutive 2's is 3, with each run of 2's separated by a run of two 1's; (3) replace each 3 in A080426 with 2. - John W. Layman (layman(AT)math.vt.edu), Feb 18 2003
a(1)=1, for n>1 a(n)= A003159(n)-A003159(n-1) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 31 2003
G.f.: Product{k>=1, 1+x^A001045(k)} - Paul Barry (pbarry(AT)wit.ie), Dec 09 2004
|
|
|
MAPLE
| ## period-doubling routine:
double:=proc(SS)
NEW:=[op(S), op(S)]:
if op(nops(NEW), NEW)=1
then NEW:=[seq(op(j, NEW), j=1..nops(NEW)-2), op(nops(NEW)-1, NEW)+1]:
else NEW:=[seq(op(j, NEW), j=1..nops(NEW)-1), op(nops(NEW)-1, NEW)-1, 1]:
fi:
end proc:
# 10 loops of the above routine generate the first 1365 terms of the sequence
# (except for the initial term):
S:=[2]:
for j from 1 to 10 do S:=double(S); od:
S;
# From Carlo Carminati, Feb 25 2011.
|
|
|
MATHEMATICA
| Length /@ Split@ Nest[ Flatten@ Join[#, # /. {1 -> 2, 2 -> 1}] &, {1}, 7]
|
|
|
CROSSREFS
| Cf. A101615.
Sequence in context: A023191 A029256 A109073 * A051486 A081355 A060778
Adjacent sequences: A026462 A026463 A026464 * A026466 A026467 A026468
|
|
|
KEYWORD
| nonn,eigen
|
|
|
AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
|
|
|
EXTENSIONS
| Corrected and extended by John W. Layman (layman(AT)math.vt.edu), Feb 18 2003
|
| |
|
|
