A006337 An "eta-sequence": a(n) = floor( (n+1)*sqrt(2) ) - floor( n*sqrt(2) ). (Formerly M0086)

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

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 2-sqrt(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 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.

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

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(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 Deléham, May 29 2006

a(n) = A159684(n-1) + 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}]] (* Jean-François Alcover, Apr 06 2012 *)

Prog

(PARI) a(n)=sqrt(2)*(n+1)\1-sqrt(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 !! (n-1)
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
(Python)
from math import isqrt
def A006337(n): return -isqrt(m:=n*n<<1)+isqrt(m+(n<<2)+2) # Chai Wah Wu, Aug 03 2022

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