A111124 -id:A111124 - cp's OEIS frontend

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

2, 22, 2211, 221121, 221121221, 22112122122112, 2211212212211211221211, 221121221221121122121121221121121, 2211212212211211221211212211211212212211212212112

Offset: 0

N. J. A. Sloane, May 07 2000

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

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

Cf. A000002, A054348, A054350, A054351, A111123, A111124.

Word lengths give A042942.

More terms from David Wasserman, Mar 04 2002

A111123 Number of 2's in n-th "Kolakoski" string defined in A054349.

Original entry on oeis.org

1, 2, 2, 3, 5, 8, 11, 16, 25, 38, 57, 85, 127, 192, 289, 430, 644, 966, 1450, 2173, 3263, 4899, 7341, 11022, 16526, 24802, 37201, 55808, 83702, 125541, 188301, 282444, 423683, 635569, 953356, 1429969, 2144990, 3217454, 4826176, 7239129, 10858479, 16287972, 24431890

Offset: 0

Benoit Cloitre, Oct 16 2005

nonn

Also the number of terms in n-th string (starting from n=3) when representing A000002 as a tree. Each branch of this tree is a string. Starting from n=3, each 1 in n-th string generates either 1 or 2 in (n+1)-th string and each 2 in n-th string generates either 11 or 22 in (n+1)-th string based on the previously generated term of either 2 or 1. Hence, the number of terms in (n+1)-th string is the sum of all terms in n-th string. - Rakesh Khanna A, May 24 2020

Rakesh Khanna A, Table of n, a(n) for n = 0..59

Cf. A001083, A042942, A054349, A111124 (number of 1's).

l = { (*terms in A042942*) }; For[i = 2, i <= Length[l], i++, Print[l[[i]] - l[[i - 1]]]]

a(0) + a(1) + ... + a(n) = A042942(n+2) - 1.

a(n) = A001083(n+4) - A001083(n+3). - Benoit Cloitre, Nov 07 2010

More terms from and offset changed to 0 by Jinyuan Wang, Apr 03 2020

cp's OEIS Frontend

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

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