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

Author

Lee Burnette, Feb 12 2016

Keywords

• nonn

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.

Examples

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

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. [dead link]
• Lee Burnette, Oscillator for n=10. [dead link]
• Stack Exchange Network chat, Initial message.
• Stack Exchange Network chat, Electrons in a wire.

Crossrefs

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

Seems to be a shifted bisection of A334504 and A334507.

Programs

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 *)

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

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.