A001030 Fixed under 1 -> 21, 2 -> 211.

2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

If treated as the terms of a continued fraction, it converges to approximately

2.57737020881617828717350576260723346479894963737498275232531856357441\

7024804797827856956758619431996. - Peter Bertok (peter(AT)bertok.com), Nov 27 2001

There are a(n) 1's between successive 2's. - Eric Angelini, Aug 19 2008

Same sequence where 1's and 2's are exchanged: A001468. - Eric Angelini, Aug 19 2008

References

• Midhat J. Gazale, Number: From Ahmes to Cantor, Section on 'Cleavages' in Chapter 6, Princeton University Press, Princeton, NJ 2000, pp. 203-211.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=1..8119
• N. G. de Bruijn, Sequences of zeros and ones generated by special production rules, Indag. Math., 43 (1981), 27-37.
• D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
• D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
• D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
• A. Nagel, A self-defining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179-183.
• N. J. A. Sloane, Handwritten notes on Self-Generating Sequences, 1970 (note that A1148 has now become A005282).
• N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Crossrefs

Length of the sequence after 'n' substitution steps is given by the terms of A000129.

Equals A004641(n) + 1.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

Programs

Haskell

Following Spage's PARI program.
a001030 n = a001030_list !! (n-1)
a001030_list = [2, 1, 1, 2] ++ f [2] [2, 1, 1, 2] where
   f us vs = ws ++ f vs (vs ++ ws) where
             ws = 1 : us ++ 1 : vs
-- Reinhard Zumkeller, Aug 04 2014

Mathematica

('n' is the number of substitution steps to perform.) Nest[Flatten[ # /. {1 -> {2, 1}, 2 -> {2, 1, 1}}] &, {1}, n]
SubstitutionSystem[{1->{2,1},2->{2,1,1}},{2},{6}][[1]] (* Harvey P. Dale, Feb 15 2022 *)

PARI

/* Fast string concatenation method giving e.g. 5740 terms in 8 iterations */
a="2";b="2,1,1,2";print1(b);for(x=1,8,c=concat([",1,",a,",1,",b]);print1(c);a=b;b=concat(b,c)) \\ K. Spage, Oct 08 2009

Python

from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n-1] if n < 9 else -isqrt(m:=(n-9)*(n-9)<<1)+isqrt(m+(n-9<<2)+2) # Chai Wah Wu, Aug 25 2022

Formula

a(n) = -1 + floor(n*(1+sqrt(2))+1/sqrt(2))-floor((n-1)*(1+sqrt(2))+1/sqrt(2)). - Benoit Cloitre, Jun 26 2004. [I don't know if this is a theorem or a conjecture. - N. J. A. Sloane, May 14 2008]

This is a theorem, following from Hofstadter's Generalized Fundamental Theorem of eta-sequences on page 10 of Eta-Lore. See also de Bruijn's paper from 1981 (hint from Benoit Cloitre). - Michel Dekking, Jan 22 2017

Extensions

More terms from Peter Bertok (peter(AT)bertok.com), Nov 27 2001