login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A014578 Binary expansion of Thue constant (or Roth's constant). 16
0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
a(0)=0; to construct the sequence start with a(1)=1, then concatenate twice and change the last term 1->0 giving 1,1,0. Concatenate those 3 terms twice giving 1,1,0,1,1,0,1,1,0, change the last term 0->1 giving 1,1,0,1,1,0,1,1,1. Concatenate those 9 terms twice and change the last term 1->0, etc. - Benoit Cloitre, Feb 09 2003
It is probably my fault if this constant is misattributed. It was "computed" circa 1971 by a very simple Life pattern (as a diagonal row of blinkers), an obvious case of the (Thue-Siegel-)Roth criterion for transcendence, since the error after 3^n bits is ~2^-3^(n+1) = O(denominator^-3). I probably should have called it Roth's constant. - Bill Gosper, Mar 19 2004
a(0) = 0; then fixed point of the morphism 1->110, 0->111, starting with a(1) = 1. - Philippe Deléham, Mar 21 2004
Characteristic function of A007417, i.e., a(n) = 1 if n is in A007417 and a(n) = 0 otherwise. - Philippe Deléham, Mar 21 2004
Multiplicative with a(3^e) = (e+1)%2, a(p^e) = 1 otherwise. - David W. Wilson, Jun 10 2005
a(A145204(n)) = 0, a(A007417(n)) = 1. - Reinhard Zumkeller, Oct 04 2008
1 if the ternary representation of n has an even number of trailing zeros. - Ralf Stephan, Sep 02 2013
LINKS
Joerg Arndt, Matters Computational (The Fxtbook), section 38.2, pp.730-731
Eric Weisstein's World of Mathematics, Thue Sequence
Eric Weisstein's World of Mathematics, Thue Constant
FORMULA
a(0)=0; for n>=1, a(n)=sum(k>=0, (-1)^k*(floor(n/3^k)-floor((n-1)/3^k))). - Benoit Cloitre, Jun 03 2003
a(0)=0, a(3k)=1-a(k); a(3k+1)=a(3k+2)=1. - Benoit Cloitre, Mar 19 2004
Sum_{k=0..3^n} a(k) = A015518(n+1) = (-1)^n*A014983(n+1). - Philippe Deléham, Mar 31 2004
a(n) = 1 - A007949(n) mod 2 for n>0. - Reinhard Zumkeller, Oct 04 2008
Let T(x) be the g.f., then T(x) + T(x^3) = x/(1-x). - Joerg Arndt, May 11 2010
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/4. - Amiram Eldar, Jul 13 2020
EXAMPLE
Start: 1
Rules:
1 --> 110
0 --> 111
-------------
0: (#=1)
1
1: (#=3)
110
2: (#=9)
110110111
3: (#=27)
110110111110110111110110110
4: (#=81)
110110111110110111110110110110110111110110111110110110110110111110110111110110111
- Joerg Arndt, Jul 06 2011
MATHEMATICA
Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 6] (* Robert G. Wilson v, Mar 09 2005 *)
PROG
(PARI) a(n)=if(n<1, 0, sum(k=0, ceil(log(n)/log(3)), (-1)^k*(floor(n/3^k)-floor((n-1)/3^k))));
(PARI) A014578(n) = if(!n, n, valuation(n, 3)%2==0); \\ Ralf Stephan, Sep 02 2013, edited for the term a(0)=0 - Antti Karttunen, May 28 2024
CROSSREFS
Cf. Thue-Morse or parity constant A010060.
Cf. A154271.
Sequence in context: A188295 A228039 A163532 * A323153 A288861 A030190
KEYWORD
nonn,cons,mult
AUTHOR
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 12 22:19 EDT 2024. Contains 374257 sequences. (Running on oeis4.)