

A054354


First differences of Kolakoski sequence A000002.


7



1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1
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OFFSET

1,1


COMMENTS

The Kolakoski sequence has only 1's and 2's, and is cubefree. Thus, for all n>=1, a(n) is in {1, 0, 1}, a(n+1) != a(n), and if a(n) = 0, a(n+1) = a(n1), while if a(n) != 0, either a(n+1) = 0 and a(n+2) = a(n) or a(n+1) = a(n). A further consequence is that the maximum gap between equal values is 4: for all n, there is an integer k, 1<k<=4 such that a(n+k)=a(n).  JeanChristophe Hervé, Oct 05 2014
From Daniel Forgues, Jul 07 2015: (Start)
Second differences: {1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, ...}
The sequence of first differences bounces between 1 and 1 with a slope whose absolute value is either 1 or 2. We can compress the information in the second differences into {1, 1, 2, 2, 1, 2, 1, 1, ...} since the 1 and the 1 come in pairs; which can be compressed further into {1, 1, 2, 2, 1, 2, 1, 1, ...} since the signs alternate, where we only need to know that the initial sign is negative. (End)


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000


FORMULA

Abs(a(n)) = (A000002(n)+A000002(n+1)) mod 2.  Benoit Cloitre, Nov 17 2003


MATHEMATICA

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1 + Mod[n  1, 2]}], {n, 3, 70}, {a2[[n]]}]; Differences[a2] (* JeanFrançois Alcover, Jun 18 2013 *)


PROG

(Haskell)
a054354 n = a054354_list !! (n1)
a054354_list = zipWith () (tail a000002_list) a000002_list
 Reinhard Zumkeller, Aug 03 2013


CROSSREFS

Cf. A000002, A054353.
Sequence in context: A123110 A004593 A094934 * A156728 A074332 A152065
Adjacent sequences: A054351 A054352 A054353 * A054355 A054356 A054357


KEYWORD

sign


AUTHOR

N. J. A. Sloane, May 07 2000


STATUS

approved



