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

Offset: 1

Author

D. R. Hofstadter, Jul 15 1977

Keywords

• nonn
• easy
• nice

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

Also, let S be the increasing sequence of elements of the union N U N*sqrt(2), where N = {1, 2, 3, ...}. Then a(n) = { 1 if S(n) is integer, 2 if S(n) is irrational }. See A245222 for the analog with sqrt(3). - M. F. Hasler, Feb 06 2025

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
• Clark 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)

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

Cf. A245222 (an analog with sqrt(3) instead of sqrt(2)).

Programs

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

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 *)

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

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

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