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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Oct 03 2019

Keywords

• nonn

Comments

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

Examples

			The evolution of one-dimensional cellular automaton rule 30 proceeds as follows, when started from a single alive (1) cell:
   0:              (1)
   1:             1(1)1
   2:            11(0)01
   3:           110(1)111
   4:          1100(1)0001
   5:         11011(1)10111
   6:        110010(0)001001
   7:       1101111(0)0111111
   8:      11001000(1)11000001
   9:     110111101(1)001000111
  10:    1100100001(0)1111011001
  11:   11011110011(0)10000101111
  12:  110010001110(0)110011010001
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, ...
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..100000
• Index entries for sequences related to cellular automata

Crossrefs

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

Programs

Mathematica

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

PARI

up_to = 105;
A269160(n) = bitxor(n, bitor(2*n, 4*n));
A327980list(up_to) = { my(v=vector(up_to), s=25, n=2, on=n, k=0); while(kA269160(s); if(!((s>>n)%2), k++; v[k] = (n-on); on=n)); (v); }
v327980 = A327980list(up_to);
A327980(n) = v327980[n];

Formula

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