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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
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%I #18 Nov 29 2019 21:19:28

%S 0,1,2,3,5,8,12,19,29,44,66,100,151,227,341,512,769,1154,1729,2591,

%T 3886,5827,8743,13117,19675,29515,44281,66432,99668,149532,224307,

%U 336451,504649,756962,1135451,1703198,2554847,3832293,5748475,8622647

%N 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

%C 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:

%C 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)

%C 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)

%C 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:

%C 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,...

%C 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,...

%C 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,...

%C 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.

%e In this Kolakoski 4-chain, seq(n-1) differs from seq(n) at the 3rd term and a(4) = 3:

%e 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,...

%e 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,...

%e 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,...

%e 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,...

%e In this Kolakoski 5-chain, seq(n-1) differs from seq(n) at the 5th term and a(5) = 5:

%e 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,...

%e 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,...

%e 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,...

%e 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,...

%e 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,...

%e In this Kolakoski 8-chain, seq(n-1) differs from seq(n) at the 19th term and a(8) = 19:

%e 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,...

%e 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,...

%e [...]

%e 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,...

%e 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,...

%Y Cf. A000002, A025142, A025143.

%K nonn,more

%O 1,3

%A _Anthony Sand_, Nov 29 2019