

A001030


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


22



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
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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. 203211.
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), 2737.
D. R. Hofstadter, EtaLore [Cached copy, with permission]
D. R. Hofstadter, PiMu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
A. Nagel, A selfdefining infinite sequence, with an application to Markoff chains and probability, Math. Mag., 36 (1963), 179183.
N. J. A. Sloane, Handwritten notes on SelfGenerating 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((n1)*(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 etasequences on page 10 of EtaLore. 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]


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 !! (n1)
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


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



