A054353 Partial sums of Kolakoski sequence A000002.

1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 17, 19, 20, 21, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 37, 39, 41, 42, 43, 45, 46, 47, 49, 50, 52, 54, 55, 57, 59, 60, 61, 63, 64, 66, 68, 69, 71, 72, 73, 75, 76, 77, 79, 81, 82, 84, 86, 87, 88, 90, 91, 93, 95, 96, 98, 100

Offset

1,2

Comments

Alternate definition: n such that A000002(n) is different from A000002(n+1). - Nathaniel Johnston, May 02 2011

Links

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

O. Bordelles and B. Cloitre, Bounds for the Kolakoski Sequence, J. Integer Sequences, 14 (2011), #11.2.1.

Bertran Steinsky, A Recursive Formula for the Kolakoski Sequence A000002, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.

Formula

A000002(a(n)) = (3+(-1)^n)/2; A000002(a(n)+1)=(3-(-1)^n)/2. - Benoit Cloitre, Oct 16 2005

a(n) = n + A074286(n) = 2*n - A156077(n) = A156077(n) + 2*A074286(n). - Jean-Christophe Hervé, Oct 05 2014

Mathematica

a2 = {1, 2, 2}; Do[ a2 = Join[a2, {1+Mod[n-1, 2]}], {n, 3, 50}, {a2[[n]] } ]; Accumulate[a2] (* Jean-François Alcover, Jun 18 2013 *)

Prog

(Haskell)
a054353 n = a054353_list !! (n-1)
a054353_list = scanl1 (+) a000002_list
-- Reinhard Zumkeller, Aug 03 2013
(Python)
from itertools import accumulate
def alst(nn):
  K = Kolakoski() # using Kolakoski() in A000002
  return list(accumulate(next(K) for i in range(1, nn+1)))
print(alst(66))   # Michael S. Branicky, Jan 12 2021

Crossrefs

Cf. A000002, A074272, A074286, A074288, A078649, A156077.

Cf. A088568 (partial sums of [3 - 2*A000002(n)]).

Sequence in context: A238246 A330998 A099467 * A284555 A031948 A247523

Adjacent sequences: A054350 A054351 A054352 * A054354 A054355 A054356

Keyword

nonn,easy

Author

N. J. A. Sloane, May 07 2000

Status

approved