

A006337


An "etasequence": floor( (n+1)*sqrt(2) )  floor( n*sqrt(2) ).
(Formerly M0086)


33



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
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OFFSET

1,2


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 (with a different offset).  Philippe Deléham, 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 2sqrt(2).  Vladimir Shevelev, Aug 05 2011
a(n) = number of repeating n's in A049472.  Reinhard Zumkeller, Jul 03 2015
Fixed point of the morphism 1 > 12; 2 > 121.  Jeffrey Shallit, Jan 19 2017


REFERENCES

Douglas Hofstadter, "Fluid Concepts and Creative Analogies", Chapter 1: "To seek whence cometh a 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..10000
F. M. Dekking, On the structure of selfgenerating sequences, Seminar on Number Theory, 19801981 (Talence, 19801981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
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
C. Kimberling, Problem 6281, Amer. Math. Monthly 86 (1979), no. 9, 793.
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)


FORMULA

Let S(0) = 1; obtain S(k) from S(k1) 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 Deléham, May 29 2006
a(n) = A159684(n1) + 1.  Filip Zaludek, Oct 28 2016


MAPLE

Digits := 100; sq2 := sqrt(2.); A006337 := n>floor((n+1)*sq2)floor(n*sq2);


MATHEMATICA

Flatten[ Table[ Nest[ Flatten[ # /. {1 > {1, 2}, 2 > {1, 1, 2}}] &, {1}, n], {n, 5}]] (* Robert G. Wilson v, May 06 2005 *)
Differences[ Table[ Floor[ n*Sqrt[2]], {n, 1, 106}]] (* JeanFrançois Alcover, Apr 06 2012 *)


PROG

(PARI) a(n)=sqrt(2)*(n+1)\1sqrt(2)*n\1 \\ Charles R Greathouse IV, Apr 06 2012
(PARI) a(n)=sqrtint(2*n^2+4*n+2)sqrtint(2*n^2) \\ Charles R Greathouse IV, Apr 06 2012
(Haskell)
a006337 n = a006337_list !! (n1)
a006337_list = f [1] where
f xs = ys ++ f ys where
ys = concatMap (\z > if z == 1 then [1, 2] else [1, 1, 2]) xs
 Reinhard Zumkeller, May 06 2012


CROSSREFS

Cf. A006338. Exchanging 1's and 2's gives A080763. Essentially same as A004641 + 1.
Cf. A049472.
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A003151 as the parent: A003151, A001951, A001952, A003152, A006337, A080763, A082844 (conjectured), A097509, A159684, A188037, A245219 (conjectured), A276862.  N. J. A. Sloane, Mar 09 2021
Sequence in context: A214364 A175922 A214856 * A214848 A006338 A020903
Adjacent sequences: A006334 A006335 A006336 * A006338 A006339 A006340


KEYWORD

nonn,easy,nice


AUTHOR

D. R. Hofstadter, Jul 15 1977


STATUS

approved



