
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 widthn strip, which is exactly what A268754 emulates. Since the initial 2 X 4n block used in this sequence corresponds to the onecell "seed" used in A268754, the resulting patterns must have the same period. (End)


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 *)
