OFFSET
0,1
COMMENTS
Fixed point of morphism 0 -> 0,1; 1 -> 0,1,0.
This sequence corresponds to the case k = 1 of the Sturmian word S_k(infinity) as defined in A080764. See A171588 for the case k = 2. - Peter Bala, Nov 22 2013
This sequence is the {1->01}-transform of the Sturmian word A080764. - Clark Kimberling, May 17 2017
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, University of Waterloo Technical Report CS-91-72, 1991. See Example 1.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms. See Example 1. [Cached copy, with permission]
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Wikipedia, Sturmian word
FORMULA
From Peter Bala, Nov 22 2013: (Start)
a(n) = floor((n + 2)*(sqrt(2) - 1)) - floor((n + 1)*(sqrt(2) - 1)).
If we read the sequence as the decimal constant C = 0.01010 01010 01010 10010 10010 ... then C = sum {n >= 1} 1/10^floor(n*(1 + sqrt(2))).
The real number 9*C has the simple continued fraction expansion [0; 11, 1010, 10000100, 100000000000100000, 100000000000000000000000000001000000000000, ...], the partial quotients having the form 10^Pell(n)*(1 + 10^Pell(n+1)) = 10^A001333(n+1) + 10^A000129(n) (see Adams and Davison).
A rapidly converging series for C is C = 9*sum {n >= 1} 10^Pell(2*n-1)*(1 + 10^Pell(2*n))/( (10^Pell(2*n-1) - 1)*(10^Pell(2*n+1) - 1) ): for example, the first 10 terms of the series give a rational approximation to C accurate to more than 130 million decimal places. Compare with the Fibonacci words A005614 and A221150. (End)
EXAMPLE
0 -> 0,1 -> 0,1,0,1,0 -> 0,1,0,1,0,0,1,0,1,0,0,1 ->...
MATHEMATICA
Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {0, 1, 0}}] &, {1}, 6] (* Robert G. Wilson v, May 02 2009 *)
SubstitutionSystem[{0->{0, 1}, 1->{0, 1, 0}}, {1}, {6}][[1]] (* Harvey P. Dale, Dec 25 2021 *)
PROG
(Haskell)
a159684 n = a159684_list !! n
a159684_list = 0 : concat (iterate (concatMap s) [1])
where s 0 = [0, 1]; s 1 = [0, 1, 0]
-- Reinhard Zumkeller, Oct 26 2013
(Python)
def aupto(nn):
Snm1, Sn = [0], [0, 1]
while len(Sn) < nn+1: Snm1, Sn = Sn, Sn + Sn + Snm1
return Sn[:nn+1]
print(aupto(104)) # Michael S. Branicky, Jul 23 2022
(Python)
from math import isqrt
def A159684(n): return -isqrt(m:=(n+1)**2<<1)+isqrt(m+(n<<2)+6)-1 # Chai Wah Wu, Aug 03 2022
CROSSREFS
See A188037 for another version of this sequence. - N. J. A. Sloane, Mar 22 2011
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862. - N. J. A. Sloane, Mar 09 2021
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Apr 19 2009
EXTENSIONS
More terms from Robert G. Wilson v, May 02 2009
STATUS
approved