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 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 (list; graph; refs; listen; history; text; internal format)
 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(%)

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Last modified December 1 20:47 EST 2023. Contains 367502 sequences. (Running on oeis4.)