cp's OEIS Frontend

A327980 Distances between successive zeros in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.

%I A327980 #15 Jun 26 2023 14:35:22
%S A327980 4,1,3,1,1,2,3,1,2,1,4,2,4,1,4,2,2,3,1,1,1,3,1,2,2,3,2,2,7,1,1,1,5,1,
%T A327980 1,2,2,4,1,1,1,1,2,1,2,3,1,1,4,1,1,3,3,3,2,1,1,1,1,1,1,2,1,1,6,4,2,1,
%U A327980 4,1,1,4,2,4,1,1,1,1,3,1,2,1,1,3,1,5,1,7,1,1,1,1,1,8,3,1,2,3,4,1,1,1,1
%N A327980 Distances between successive zeros in A051023, the middle column of rule-30 1-D cellular automaton, when started from a lone 1 cell.
%C A327980 First differences of A327985, which gives indices of zeros in A051023.
%H A327980 Antti Karttunen, <a href="/A327980/b327980.txt">Table of n, a(n) for n = 1..100000</a>
%H A327980 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A327980 a(n) = A327985(1+n) - A327985(n).
%e A327980 The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
%e A327980    0:              (1)
%e A327980    1:             1(1)1
%e A327980    2:            11(0)01
%e A327980    3:           110(1)111
%e A327980    4:          1100(1)0001
%e A327980    5:         11011(1)10111
%e A327980    6:        110010(0)001001
%e A327980    7:       1101111(0)0111111
%e A327980    8:      11001000(1)11000001
%e A327980    9:     110111101(1)001000111
%e A327980   10:    1100100001(0)1111011001
%e A327980   11:   11011110011(0)10000101111
%e A327980   12:  110010001110(0)110011010001
%e A327980 When noting up the distances between successive 0's in its central column (indicated here with parentheses), we obtain 6-2 (as the first 0 is on row 2, and the second is on row 6), 7-6, 10-7, 11-10, 12-11, ..., that is, the first terms of this sequence: 4, 1, 3, 1, 1, ...
%t A327980 A327980list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],0]]];A327980list[300] (* _Paolo Xausa_, Jun 01 2023 *)
%o A327980 (PARI)
%o A327980 up_to = 105;
%o A327980 A269160(n) = bitxor(n, bitor(2*n, 4*n));
%o A327980 A327980list(up_to) = { my(v=vector(up_to), s=25, n=2, on=n, k=0); while(k<up_to, n++; s = A269160(s); if(!((s>>n)%2), k++; v[k] = (n-on); on=n)); (v); }
%o A327980 v327980 = A327980list(up_to);
%o A327980 A327980(n) = v327980[n];
%Y A327980 Cf. A051023, A110240, A245549, A269160, A327981, A327983, A327985.
%K A327980 nonn
%O A327980 1,1
%A A327980 _Antti Karttunen_, Oct 03 2019