

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


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

D. R. Hofstadter, EtaLore [Cached copy, with permission]


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]
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 !! (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
(Python)
from math import isqrt
def A001030(n): return [2, 1, 1, 2, 1, 2, 1, 2][n1] if n < 9 else isqrt(m:=(n9)*(n9)<<1)+isqrt(m+(n9<<2)+2) # Chai Wah Wu, Aug 25 2022


CROSSREFS

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


KEYWORD

nonn,nice,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



