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%I #31 Apr 11 2018 02:59:19
%S 1,1,1,3,1,3,1,1,3,1,1,3,1,1,3,1,1,1,3,2,1,1,3,2,1,3,1,1,3,1,1,1,3,2,
%T 1,1,3,2,1,1,3,2,1,1,3,3,1,1,3,1,2,2,1,1,3,1,2,1,1,3,1,1,3,1,1,1,3,2,
%U 1,1,3,2,1,1,3,2,1,1,3,3,1,1,3,1,2,2,1
%N A "Look and Say" sequence (with nested repetitions).
%C A variant of the 'Look-and-Say' sequence A005150 that describes at each step the preceding digits altogether since the beginning. The sequence is built by blocks, each new block describing the preceding ones, always returning to the beginning: 1; 1,1; 3,1; 3,1,1,3,1,1; 3,1,1,3,1,1,1,3,2,1,1,3,2,1; etc. This generates indefinitely nested repeats: 31, 311311, 311311, ... The sequence A225224 is a variant that avoids these repetitions.
%C The size of the block of rank k equals 2^k for k = 1 or 2, and is < 2^k for any k >= 3.
%C Except the first two blocks, each block begins with 31 and ends with 1.
%C As in the original A005150, a(n) is always equal to 1, 2 or 3. The subsequence 3,3,3 never appears.
%H John Cerkan, <a href="/A225212/b225212.txt">Table of n, a(n) for n = 1..10000</a>
%e a(1) = 1, you then see "one 1": a(2) = 1 (one) and a(3) = 1; Looking at a(1), a(2) and a(3) altogether, you then see "three 1" : a(4) = 3 and a(5) = 1. The sequence is then built by blocks, each new block describing the preceding ones since the beginning of the sequence: 1; 1,1; 3,1; 3,1,1,3,1,1; etc.
%t f[seq_] := Join[seq, {Length[#], First[#]}& /@ Split[seq]] // Flatten; Nest[f, {1}, 5] (* _Jean-François Alcover_, May 02 2013 *)
%Y Cf. A005150, A225224.
%K nonn,easy
%O 1,4
%A _Jean-Christophe Hervé_, May 01 2013
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