A180001 Eventual period of a single cell in rule 110 cellular automaton in a cyclic universe of width n.

1, 1, 1, 2, 1, 9, 14, 16, 7, 25, 110, 9, 351, 91, 295, 32, 7, 27, 285, 30, 630, 44, 1058, 36, 250, 7, 405, 1652, 1044, 60, 7, 64, 495, 51, 1050, 72, 4403, 76, 390, 60, 7, 630, 1548, 88, 7, 7, 705, 96, 1470, 100, 765, 195, 8109, 7, 825, 7, 2052, 116, 7, 19560, 915

Offset

1,4

Comments

The first 21 terms match the most frequent possible outcome (see comment in A332717) with the exception of a(14) which is the second-most frequent. - Hans Havermann, Jun 11 2020

Links

Pontus von Brömssen, Table of n, a(n) for n = 1..1000

Eric Weisstein's World of Mathematics, Rule 110

Index entries for sequences related to cellular automata

Example

For n=4, the evolution of a single cell is:
0001
0011
0111 <--= period starts
1101
0111 <--= again start of period
etc, so a(4)=2.

Mathematica

a[n_] := -Subtract @@
   Flatten[Map[Position[#, #[[-1]]] &,
     NestWhileList[CellularAutomaton[110],
      Prepend[Table[0, {n - 1}], 1], Unequal, All], {0}]]

Prog

(Sage)
def A180001(n):
    def bit(x, i): return (x >> i) & 1
    rulemap = dict((tuple(bit(i, k) for k in reversed(range(3))), bit(110, i)) for i in range(8))
    def neighbours(d, i): return tuple(d[k % n] for k in [i-1..i+1])
    v = [0]*n; v[-1] = 1;
    history = [v]
    while True:
        v2 = [rulemap[neighbours(history[-1], i)] for i in range(n)]
        if v2 in history: return len(history)-history.index(v2)
        history.append(v2) # D. S. McNeil, Jan 15 2011

Crossrefs

Cf. A085587, A334496, A334497, A332717, A334499-A334515.

Sequence in context: A085488 A072265 A192352 * A204371 A199887 A058876

Adjacent sequences: A179998 A179999 A180000 * A180002 A180003 A180004

Keyword

nonn

Author

Ben Branman, Jan 13 2011

Extensions

More terms from Alois P. Heinz, Jan 14 2011

Status

approved