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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 (list; graph; refs; listen; history; text; internal format)
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(n-1), 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). - Jean-Christophe 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] (* Jean-François Alcover, Jun 18 2013 *)

PROG

(Haskell)

a054354 n = a054354_list !! (n-1)

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

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Last modified July 28 20:52 EDT 2015. Contains 260095 sequences.