A160657 a(n) is the period of a 2 X 4n rectangular oscillator in the 2 X 2 (B36/S125) Life-like cellular automaton.

2, 6, 14, 14, 62, 126, 30, 30, 1022, 126, 4094, 2046, 1022, 32766, 62, 62, 8190, 174762, 8190, 2046, 254, 8190, 16777214, 4194302, 510, 134217726, 2097150, 1022, 1073741822, 2147483646, 126, 126, 17179869182, 8388606, 68719476734, 1022, 2097150, 2147483646

Offset: 1

Author

Nathaniel Johnston, May 22 2009

Keywords

• nonn

Comments

These oscillators work and have the same period in any rule from B3/S5 to B3678/S012567.

The Nathaniel Johnston rectangular oscillator link points to Sierpinski's gasket (Pascal's triangle mod 2) as a source for the chaotic terms of A003558. This is consistent with the comment of [Sep 21 2011, A003558] showing an alternative trigonometric connection to A054142, since the latter row terms are found as alternate ascending diagonals in Pascal's triangle. - Gary W. Adamson, Sep 21 2011

From Charlie Neder, Jan 11 2019: (Start)

a(n) = A268754(2n).

Proof: Decompose the phases of the oscillators into rectangles, as in the linked paper. Each of these rectangles has a corner on the exterior of the bounding diamond of the oscillator which determines the rectangle. As shown in the paper, these corners behave as Rule 90 on a width-n strip, which is exactly what A268754 emulates. Since the initial 2 X 4n block used in this sequence corresponds to the one-cell "seed" used in A268754, the resulting patterns must have the same period. (End)

Examples

			a(2) = 6 because a 2 X 8 box has period 6 in this rule.
		

Links

• Adam P. Goucher, Table of n, a(n) for n = 1..160
• Nathaniel Johnston, Rectangular Oscillators in the 2*2 (B36/S125) Cellular Automaton, 2009.
• Nathaniel Johnston, The B36/S125 "2×2" Life-Like Cellular Automaton, arXiv:1203.1644 [nlin.CG], 2012; also in Game of Life Cellular Automata, A. Adamatzky (ed.), Springer-UK, 2010, pages 99-114.
• LifeWiki, 2x2

Crossrefs

Cf. A003558, A054142, A268754, A298819.

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[2 n], vec[2 n], 2], {n, 1, 100}]
(* Adam P. Goucher, Jan 13 2019 *)

Formula

a(n) divides 2^(A003558(n) + 1) - 2 for n >= 1. [Corrected by Charlie Neder, Jan 11 2019]

Extensions

a(18) corrected by Charlie Neder, Jan 11 2019