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A160657 a(n) is the period of a 2 X 4n rectangular oscillator in the 2 X 2 (B36/S125) Life-like cellular automaton. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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)

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

FORMULA

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

EXAMPLE

a(2) = 6 because a 2 X 8 box has period 6 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[2 n], vec[2 n], 2], {n, 1, 100}]

(* Adam P. Goucher, Jan 13 2019 *)

CROSSREFS

Cf. A003558, A054142, A268754, A298819.

Sequence in context: A084106 A295987 A263691 * A222087 A293654 A128660

Adjacent sequences:  A160654 A160655 A160656 * A160658 A160659 A160660

KEYWORD

nonn

AUTHOR

Nathaniel Johnston, May 22 2009

EXTENSIONS

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

STATUS

approved

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Last modified February 21 06:40 EST 2019. Contains 320371 sequences. (Running on oeis4.)