A268754 The period of an n X 1 rectangular oscillator in the B1/S Life-like cellular automaton.

1, 2, 1, 6, 4, 14, 1, 14, 12, 62, 8, 126, 28, 30, 1, 30, 28, 1022, 24, 126, 124, 4094, 16, 2046, 252, 1022, 56, 32766, 60, 62, 1, 62, 60, 8190, 56, 174762, 2044, 8190, 48, 2046, 252, 254, 248, 8190, 8188, 16777214, 32, 4194302, 4092, 510, 504, 134217726, 2044, 2097150

Offset

1,2

Comments

The seed in each case is a single live cell at the left end.

Terms of the form 2^k-1 have a period of 1 since all cells die.

In binary, all terms (except the 1's) have at least one 1 followed by at least one 0. The exceptions are the 36th and 94th terms and their derivatives, which have alternating 1's and 0's in their binary expansion.

Links

Adam P. Goucher, Table of n, a(n) for n = 1..200 (terms for n = 1..99 from E-Hern Lee)

Lee Burnette, Variations of Life.

Lee Burnette, Oscillator for n=10.

Stack Exchange Network chat, Initial message.

Stack Exchange Network chat, Electrons in a wire.

Formula

No general formula for even-indexed terms is known. For odd-indexed terms, a(2n+1) = 2*a(n), except when n is of the form (2^k - 1), in which case a(n) = 1.

Example

a(10) = 62 because a strip of 10 cells has period 62 in this rule.

Mathematica

g = Function[{sq, p}, Module[{l = Length[sq]},
Do[If[sq[[i]] == sq[[j]], Return[p^(j - 1) - p^(i - 1)]],
{j, 2, l}, {i, 1, j - 1}]]];
MPM = Algebra`MatrixPowerMod;
EventualPeriod = Function[{m, v, p},
Module[{n = Length[m], w, sq, k, primes},
sq = NestList[(MPM[#, p, p]) &, m, n];
w = Mod[Last[sq].v, p];
sq = Map[(Mod[#.w, p]) &, sq];
k = g[sq, p];
If[k == Null, k = p^n Apply[LCM, Table[p^r - 1, {r, 1, n}]]];
primes = Map[First, FactorInteger[k]];
primes = Select[primes, (# > 1) &];
While[Length[primes] > 0,
primes = Select[primes, (Mod[k, #] == 0) &];
primes = Select[primes, (Mod[MPM[m, k/#, p].w, p] == w) &];
k = k/Fold[Times, 1, primes];
]; k ]];
mat = Function[{n}, Table[Boole[Abs[i - j] == 1], {i, 1, n}, {j, 1, n}]];
vec = Function[{n}, Table[Boole[i == 1], {i, 1, n}]];
Table[EventualPeriod[mat[n], vec[n], 2], {n, 1, 100}]
(* Adam P. Goucher, Jan 13 2019 *)

Prog

(Python)
def electron_period(n):
  wire_mask = (1 << n) - 1
  power = lam = 1
  tortoise, hare = 1, 2
  while tortoise != hare:
    if power == lam:
      tortoise = hare
      power *= 2
      lam = 0
    hare = ((hare << 1) ^ (hare >> 1)) & wire_mask
    lam += 1
  return lam

Crossrefs

Even-indexed terms are exactly A160657. [corrected by Adam P. Goucher, Jan 13 2019]

Sequence in context: A121403 A155550 A355642 * A357639 A005299 A185586

Adjacent sequences: A268751 A268752 A268753 * A268755 A268756 A268757

Keyword

nonn

Author

Lee Burnette, Feb 12 2016

Status

approved