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

Author

Ben Branman, Jan 13 2011

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

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

Programs

Mathematica

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

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

Extensions

More terms from Alois P. Heinz, Jan 14 2011