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Run lengths in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.
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%I #10 Oct 04 2019 15:05:54

%S 2,1,3,2,2,3,1,1,2,2,1,2,3,1,1,1,3,2,3,1,1,1,1,1,2,4,2,2,1,1,1,1,2,1,

%T 1,1,1,1,6,4,4,3,1,1,1,1,3,5,1,2,1,1,2,3,3,3,2,1,2,1,2,1,1,7,1,3,5,1,

%U 3,1,1,2,3,3,3,1,1,1,3,5,2,2,1,3,2,2,4,2,6,6,7,1,2,2,1,1,2,1,3,5,1,1,2,3,2

%N Run lengths in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

%H Antti Karttunen, <a href="/A327983/b327983.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%e The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:

%e 0: (1)

%e 1: 1(1)1

%e 2: 11(0)01

%e 3: 110(1)111

%e 4: 1100(1)0001

%e 5: 11011(1)10111

%e 6: 110010(0)001001

%e 7: 1101111(0)0111111

%e 8: 11001000(1)11000001

%e 9: 110111101(1)001000111

%e 10: 1100100001(0)1111011001

%e 11: 11011110011(0)10000101111

%e 12: 110010001110(0)110011010001

%e 13: 1101111011001(1)1011100110111

%e When noting up the lengths of consecutive identical values ("runs") in its central column (indicated here with parentheses), we see that there are two ones at first, followed by one zero, followed by three ones, then two zeros, etc, and so we obtain the terms on this sequence: 2, 1, 3, 2, 2, 3, ...

%t Length /@ Split@ CellularAutomaton[30, {{1}, 0}, {105, {{0}}}] (* _Michael De Vlieger_, Oct 04 2019 *)

%o (PARI)

%o up_to = 105;

%o A269160(n) = bitxor(n, bitor(2*n, 4*n));

%o A327983list(up_to) = { my(v=vector(up_to), s=1, oc=s, nc, n=0, on=n, k=0); while(k<up_to, n++; s = A269160(s); nc = (s>>n)%2; if(nc!=oc, oc=nc; k++; v[k] = (n-on); on=n)); (v); }

%o v327983 = A327983list(up_to);

%o A327983(n) = v327983[n];

%Y Cf. A051023, A269160, A327974, A327980, A327981, A327984, A327985.

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 03 2019