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A006337
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An "eta-sequence": [ (n+1)*sqrt(2) ] - [ n*sqrt(2) ].
(Formerly M0086)
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9
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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, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Defined by: (i) a(1) = 1; (ii) sequence consists of single 2's separated by strings of 1's; (iii) the sequence of lengths of runs of 1's in the sequence is equal to the sequence.
Equals its own "derivative", which is formed by counting the strings of 1's that lie between 2's.
First differences of A001951. - Philippe DELEHAM, May 29 2006
Or number of perfect squares in interval (2*n^2, 2*(n+1)^2). In view of the uniform distribution mod 1 of sequence {sqrt(2)*n}, the density of 1's is 2-sqrt(2). -Vladimir Shevelev, Aug 05 2011
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REFERENCES
| F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a sequence".
C. Kimberling, Problem 6281, Amer. Math. Monthly 86 (1979), no. 9, 793.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
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FORMULA
| Let S(0) = 1; obtain S(k) from S(k-1) by applying 1 -> 12, 2 -> 112; sequence is S(0), S(1), S(2), ... - Matthew Vandermast, Mar 25 2003
a(A003152(n)) = 1 and a(A003151(n)) = 2 . - Philippe DELEHAM, May 29 2006
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MAPLE
| Digits := 100; sq2 := sqrt(2.); A006337 := n->floor((n+1)*sq2)-floor(n*sq2);
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MATHEMATICA
| Flatten[ Table[ Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {1, 1, 2}}] &, {1}, n], {n, 5}]] (from Robert G. Wilson v, May 06 2005)
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CROSSREFS
| Cf. A006338. Exchanging 1's and 2's gives A080763. Essentially same as A004641 + 1.
Sequence in context: A022300 A105690 A175922 * A006338 A020903 A133083
Adjacent sequences: A006334 A006335 A006336 * A006338 A006339 A006340
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KEYWORD
| nonn,easy,nice
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AUTHOR
| D. R. Hofstadter
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