cp's OEIS Frontend

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

%I A180001 #41 Feb 16 2025 08:33:12
%S A180001 1,1,1,2,1,9,14,16,7,25,110,9,351,91,295,32,7,27,285,30,630,44,1058,
%T A180001 36,250,7,405,1652,1044,60,7,64,495,51,1050,72,4403,76,390,60,7,630,
%U A180001 1548,88,7,7,705,96,1470,100,765,195,8109,7,825,7,2052,116,7,19560,915
%N A180001 Eventual period of a single cell in rule 110 cellular automaton in a cyclic universe of width n.
%C A180001 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
%H A180001 Pontus von Brömssen, <a href="/A180001/b180001.txt">Table of n, a(n) for n = 1..1000</a>
%H A180001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Rule110.html">Rule 110</a>
%H A180001 <a href="http://oeis.org/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%e A180001 For n=4, the evolution of a single cell is:
%e A180001 0001
%e A180001 0011
%e A180001 0111 <--= period starts
%e A180001 1101
%e A180001 0111 <--= again start of period
%e A180001 etc, so a(4)=2.
%t A180001 a[n_] := -Subtract @@
%t A180001    Flatten[Map[Position[#, #[[-1]]] &,
%t A180001      NestWhileList[CellularAutomaton[110],
%t A180001       Prepend[Table[0, {n - 1}], 1], Unequal, All], {0}]]
%o A180001 (Sage)
%o A180001 def A180001(n):
%o A180001     def bit(x,i): return (x >> i) & 1
%o A180001     rulemap = dict((tuple(bit(i,k) for k in reversed(range(3))), bit(110,i)) for i in range(8))
%o A180001     def neighbours(d, i): return tuple(d[k % n] for k in [i-1..i+1])
%o A180001     v = [0]*n; v[-1] = 1;
%o A180001     history = [v]
%o A180001     while True:
%o A180001         v2 = [rulemap[neighbours(history[-1], i)] for i in range(n)]
%o A180001         if v2 in history: return len(history)-history.index(v2)
%o A180001         history.append(v2) # _D. S. McNeil_, Jan 15 2011
%Y A180001 Cf. A085587, A334496, A334497, A332717, A334499-A334515.
%K A180001 nonn
%O A180001 1,4
%A A180001 _Ben Branman_, Jan 13 2011
%E A180001 More terms from _Alois P. Heinz_, Jan 14 2011