A001030 Fixed under 1 -> 21, 2 -> 211. (Formerly M0068 N0021)

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

1,1

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).

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

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 *)

Prog

A001030 := proc(n) begin [ 2 ]; while nops(%)<n do subs(%, [ 1=(2, 1), 2=(2, 1, 1) ]) end_while; %[ n ] end_proc:
(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
(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
(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

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

Sequence in context: A134265 A182858 A175077 * A246140 A071709 A131406

Adjacent sequences: A001027 A001028 A001029 * A001031 A001032 A001033

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved