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 1-3 of 3 results.

A111090 Successive generations of an alternating Kolakoski rule.

Original entry on oeis.org

1, 2, 11, 21, 112, 2122, 1121122, 2122121122, 1121122122121122, 212212112212211211212211, 112112212212112212211212212112112212
Offset: 1

Views

Author

Benoit Cloitre, Oct 12 2005

Keywords

Comments

Strings are obtained using the Kolakoski substitution and the additional rule: start with 1 if previous string begins with 2, start with 2 if previous string begins with 1.
a(n+1) > a(n), and a(n) is always composed of 1s and 2s, hence a subsequence of A007931. - Charles R Greathouse IV, Nov 20 2024

Examples

			1-->2-->11-->21-->112-->2122

Crossrefs

Subsequence of A007931.
Cf. A111081, A000002.

Formula

As n grows a(2n-1) converges toward A025142 (red as a word) and a(2n) converges toward A025143. Conjecture : a(n) is asymptotic to c*(3/2)^n for some c.

A350505 Irregular triangle read by rows: a(n) is the n-th term of a truncated Kolakoski tree.

Original entry on oeis.org

1, 2, 7, 3, 4, 15, 24, 5, 6, 14, 10, 35, 122, 36, 37, 11, 61, 8, 9, 21, 22, 34, 23, 53, 54, 184, 185, 55, 56, 84, 126, 16, 17, 136, 311, 12, 13, 20, 48, 32, 33, 113, 51, 392, 78, 52, 79, 119, 405, 80, 81, 277, 278, 414, 621, 82, 83, 283, 125, 425, 191, 2166, 288
Offset: 1

Views

Author

John-Vincent Saddic, Jan 02 2022

Keywords

Comments

Construct a tree with these rules: The root node is labeled 1. Children of odd indices are labeled 2. Children of even indices are labeled 1. Nodes labeled 1 have one child. Nodes labeled 2 have two children. At this point, the values of the n-th row comprise the digits in A111081(n). Replace each node labeled 2 with the positive integers in breadth-first order. Remove all nonbranching nodes.

Examples

			Triangle begins:
  1;
  2, 7;
  3, 4, 15, 24;
  5, 6, 14, 10, 35, 122, 36, 37;
...

Links

Crossrefs

Inverse is A350506.
Cf. A111081.

A111091 Successive generations of a Kolakoski(3,1) rule starting with 1 (see A066983).

Original entry on oeis.org

1, 3, 111, 313, 1113111, 313111313, 11131113131113111, 3131113131113111313111313, 1113111313111311131311131311131113131113111
Offset: 1

Views

Author

Benoit Cloitre, Oct 12 2005

Keywords

Comments

Terms are palindromic. If b_3(n) denotes the number of 3's in a(n) then b(n) satisfies the recursion: b_3(1)=0, b_3(2)=1 and b_3(n) = b_3(n-1) + b_3(n-2) + (-1)^n so that b_3(2n)=A055588(n) and b_3(2n+1)=A027941(n). If b_1(n) denotes the number of 1's: b_1(1)=1, b_1(2)=0 and b_1(n) = b_1(n-1) + b_1(n-2) - 2*(-1)^n so that b_1(2n)=A004146(n) and b_1(2n+1) = A000032(n-2) - 2.

Examples

			1 --> 3 --> 111 --> 313 --> 1113111 --> 313111313

Crossrefs

Cf. A111081.

Formula

As n grows, a(2n-1) converges toward A095345 (read as a word) and a(2n) converges toward A095346.
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.