A327980 - OEIS

%I #15 Jun 26 2023 14:35:22

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

%C First differences of A327985, which gives indices of zeros in A051023.

%H Antti Karttunen, <a href="/A327980/b327980.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>

%F a(n) = A327985(1+n) - A327985(n).

%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 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 A327980list[upto_]:=Differences[Flatten[Position[CellularAutomaton[30,{{1},0},{upto,{{0}}}],0]]];A327980list[300] (* _Paolo Xausa_, Jun 01 2023 *)

%o (PARI)

%o up_to = 105;

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

%o 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 v327980 = A327980list(up_to);

%o A327980(n) = v327980[n];

%Y Cf. A051023, A110240, A245549, A269160, A327981, A327983, A327985.

%K nonn

%O 1,1

%A _Antti Karttunen_, Oct 03 2019