

A006337


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


35



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.
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
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

D. R. Hofstadter, EtaLore [Cached copy, with permission]
C. Kimberling, Problem 6281, Amer. Math. Monthly 86 (1979), no. 9, 793.


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


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 *)


PROG

(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
(Python)
from math import isqrt


CROSSREFS

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


KEYWORD

nonn,easy,nice


AUTHOR

D. R. Hofstadter, Jul 15 1977


STATUS

approved



