A327421 In a Kolakoski n-chain, point at which term of penultimate sequence seq(n-1) differs from term of final sequence seq(n) in chain, when terms of seq(i) are run-lengths of seq(i+1) and the chain contains n sequences.

0, 1, 2, 3, 5, 8, 12, 19, 29, 44, 66, 100, 151, 227, 341, 512, 769, 1154, 1729, 2591, 3886, 5827, 8743, 13117, 19675, 29515, 44281, 66432, 99668, 149532, 224307, 336451, 504649, 756962, 1135451, 1703198, 2554847, 3832293, 5748475, 8622647

Offset: 1

Author

Anthony Sand, Nov 29 2019

Keywords

• nonn
• more

Comments

The terms of the Kolakoski sequence, A000002, are the run-lengths of the same sequence. The terms of the sequence never differ from themselves and a(1) is therefore assigned the value 0. In a Kolakoski n-chain consisting of n >= 2 sequences, the terms of seq(i) are the run-lengths of seq(i+1), with the final sequence, seq(n), in the chain being the run-lengths of seq(1). The sequence above, a(n), records the term at which seq(n-1) differs from seq(n) in a chain of n sequences that use the alphabets {2,1} for seq(1) and {1,2} for seq(2..n). For example, in the Kolakoski 2-chain, A025142 and A025143, the sequences are:

seq(1) = 2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,... (A025143)

seq(2) = 1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,... (A025142)

The penultimate sequence, seq(n-1 = 1), differs from the final sequence, seq(n = 2), at the 1st term and therefore a(2) = 1. In this Kolakoski 3-chain, seq(n-1) differs from seq(n) at the 2nd term and a(3) = 2:

seq(1) = 2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,1,2,...

seq(2) = 1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,...

seq(3) = 1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,...

Conjectures: 1) In a Kolakoski n-chain of the form given, as n -> infinity, seq(n) converges on the Kolakoski sequence, A000002, whose terms always match its own run-lengths, while seq(1) converges on the anti-Kolakoski sequence, A049705, whose terms never match its own run-lengths. 2) As i -> infinity, a(i) / a(i+1) converges on 2/3.

Examples

			In this Kolakoski 4-chain, seq(n-1) differs from seq(n) at the 3rd term and a(4) = 3:
seq(1) = 2,1,1,2,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,...
seq(2) = 1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,...
seq(3) = 1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,2,1,2,2,1,2,1,1,2,...
seq(4) = 1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,...
In this Kolakoski 5-chain, seq(n-1) differs from seq(n) at the 5th term and a(5) = 5:
seq(1) = 2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,...
seq(2) = 1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,...
seq(3) = 1,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,...
seq(4) = 1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,...
seq(5) = 1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,...
In this Kolakoski 8-chain, seq(n-1) differs from seq(n) at the 19th term and a(8) = 19:
seq(1) = 2,1,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,...
seq(2) = 1,1,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,...
[...]
seq(7) = 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,...
seq(8) = 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,2,2,1,1,2,...
		

Crossrefs

Cf. A000002, A025142, A025143.