cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A054351 Successive generations of the Kolakoski sequence A000002.

Original entry on oeis.org

1, 12, 1221, 1221121, 12211212212, 122112122122112112, 1221121221221121122121121221, 1221121221221121122121121221121121221221121, 12211212212211211221211212211211212212211212212112112212211212212
Offset: 0

Views

Author

N. J. A. Sloane, May 07 2000

Keywords

Crossrefs

Cf. A054348, A054349, A054350, A054352, A042942, A000002.
Word lengths give A054352.

Programs

  • Python
    from itertools import accumulate, groupby, repeat
def K(n, _):
  c, s = "12", ""
  for i, k in enumerate(str(n)): s += c[i%2]*int(k)
  return int(s + c[(i+1)%2])
def aupton(nn): return list(accumulate(repeat(1, nn+1), K))
print(aupton(8)) # Michael S. Branicky, Jan 12 2021

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 05 2003

A054352 Lengths of successive generations of the Kolakoski sequence A000002.

Original entry on oeis.org

1, 2, 4, 7, 11, 18, 28, 43, 65, 99, 150, 226, 340, 511, 768, 1153, 1728, 2590, 3885, 5826, 8742, 13116, 19674, 29514, 44280, 66431, 99667, 149531, 224306, 336450, 504648, 756961, 1135450, 1703197, 2554846, 3832292, 5748474, 8622646, 12933971, 19400955, 29101203
Offset: 0

Views

Author

N. J. A. Sloane, May 07 2000

Keywords

Comments

Starting with a(0) = 1, the first term of A000002, the n-th generation is the run of figures directly generated from the preceding generation completed with a single last figure which begins the next run. Thus a(0) = 1 -> 1-2 -> 1-22-1 -> 1-2211-2-1 etc. - Jean-Christophe Hervé, Oct 26 2014
It seems that the limit (c =) lim_{n -> oo} a(n)/(3/2)^n exists, with c = 2.63176..., so a(n) ~ (3/2)*a(n-1) ~ c * (3/2)^n, for large n. - A.H.M. Smeets, Apr 12 2024

Links

Crossrefs

Cf. A054348, A054349, A054350, A054351, A054353, A042942, A000002.
Partial sums of A329758.

Programs

  • Mathematica
    A2 = {1, 2, 2}; Do[If[Mod[n, 10^5] == 0, Print["n = ", n]]; m = 1 + Mod[n - 1, 2]; an = A2[[n]]; A2 = Join[A2, Table[m, {an}]], {n, 3, 10^6}]; A054353 = Accumulate[A2]; Clear[a]; a[0] = 1; a[n_] := a[n] = A054353[[a[n - 1]]] + 1; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Oct 30 2014, after Jean-Christophe Hervé *)
  • Python
    def aupton(nn):
  alst, A054353, idx = [1], 0, 1
  K = Kolakoski()  # using Kolakoski() in A000002
  for n in range(2, nn+1):
    target = alst[-1]
    while idx <= target:
      A054353 += next(K)
      idx += 1
    alst.append(A054353 + 1)  # a(n) = A054353(a(n-1))+1
  return alst
print(aupton(36))  # Michael S. Branicky, Jan 12 2021

Formula

a(0) = 1, and for n > 0, a(n) = A054353(a(n-1))+1. - Jean-Christophe Hervé, Oct 26 2014

Extensions

a(7)-a(32) from John W. Layman, Aug 20 2002
a(33) from Jean-François Alcover, Oct 30 2014
a(34) and beyond from Michael S. Branicky, Jan 12 2021

A054349 Successive generations of the variant of the Kolakoski sequence described in A042942.

Original entry on oeis.org

2, 22, 2211, 221121, 221121221, 22112122122112, 2211212212211211221211, 221121221221121122121121221121121, 2211212212211211221211212211211212212211212212112
Offset: 0

Views

Author

N. J. A. Sloane, May 07 2000

Keywords

Comments

For n >= 0, let f_1(n) be the number of 1's in a(n) (sequence begins: 0, 0, 2, 3, 4, 6, 11, 17, 24, ...) and f_2(n) be the number of 2's (sequence begins: 1, 2, 2, 3, 5, 8, 11, 16, 25, ...). Then there is a simple relation between f_1 and f_2, namely: f_1(n) = 1 - f_2(n) + f_2(n-1) + f_2(n-2) + ... + f_2(0). i.e. f_1(7) = 17 and 1 - f_2(7) + f_2(6) + ... + f_2(0) = 1 - 16 + 11 + 8 + 5 + 3 + 2 + 2 + 1 = 17. - Benoit Cloitre, Oct 11 2005

Links

Crossrefs

Cf. A000002, A054348, A054350, A054351, A111123, A111124.
Word lengths give A042942.

Extensions

More terms from David Wasserman, Mar 04 2002

A054348 Triangular array whose rows are successive generations of the variant of the Kolakoski sequence described in A042942.

Original entry on oeis.org

2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1
Offset: 0

Views

Author

N. J. A. Sloane, May 07 2000

Keywords

Examples

			Triangle begins:
2;
2, 2;
2, 2, 1, 1;
2, 2, 1, 1, 2, 1;
2, 2, 1, 1, 2, 1, 2, 2, 1;
2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2;
...

Crossrefs

Cf. A000002, A054349, A054350, A054351.
Row lengths give A042942.

Extensions

More terms from David Wasserman, Mar 04 2002
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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