cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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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.

Original entry on oeis.org

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

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Author

Nathaniel Johnston, May 22 2009

Keywords

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

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

A298819 a(n) is the period of the oscillating pattern formed by a 1 X n line of cells in the Life-like cellular automaton B2c3-cekq4ikt5i8/S2-in3-acky4aijry5eiky6i, or 0 if the pattern vanishes.

Original entry on oeis.org

0, 0, 2, 2, 2, 0, 6, 6, 12, 4, 28, 14, 14, 0, 14, 14, 84, 12, 372, 62, 248, 8, 504, 126, 252, 28, 420, 30, 30, 0, 30, 30, 420, 28, 2044, 1022, 12264, 24, 504, 126, 7812, 124, 253828, 4094, 32752, 16, 16368, 2046, 85932, 252, 18396, 1022, 4088, 56, 917448, 32766, 327660, 60, 1860, 62, 62, 0, 62, 62
Offset: 1

Views

Author

WG Zeist, Jan 26 2018

Keywords

Comments

An explanation of the Hensel notation used to define the cellular automaton rule can be found on the LifeWiki (see links).

Examples

			a(4) = 2 because a 1 X 4 line oscillates with period 2 in this cellular automaton.

Links

Crossrefs

Cf. A000918, A003558, A160657, A268754.

Formula

For n divisible by 4, a(n) = A160657(n/4) = (a divisor of) 2^(A003558(n/4) + 1) - 2.
With the exception of a(1), all values of n for which a(n) = 0 are of the form 2^m - 2 for some integer m (A000918).
a(2k) = A268754(k); a(2k+1) = lcm(A268754(k), A268754(k+1)) (conjectured). Note that, by an arbitrary/cosmetic convention, A268754 uses 1 for empty patterns and A298819 uses 0. - WG Zeist and Charlie Neder, Jan 11 2019

A334504 Eventual period of a single cell in rule 26 cellular automaton in a cyclic universe of width n.

Original entry on oeis.org

1, 1, 6, 1, 20, 2, 28, 1, 72, 6, 88, 4, 104, 14, 120, 1, 272, 14, 304, 12, 336, 62, 368, 8, 400, 126, 432, 28, 464, 30, 496, 1, 1056, 30, 1120, 28, 1184, 1022, 1248, 24, 1312, 126, 1376, 124, 1440, 4094, 1504, 16, 1568, 2046, 1632, 252, 1696, 1022, 1760
Offset: 1

Views

Author

N. J. A. Sloane, May 05 2020

Keywords

Comments

Bradley Klee computed a(1)-a(10).

References

  • Bradley Klee, Posting to Math Fun Mailing List, Apr 26 2020

Links

Crossrefs

Cf. A180001, A334496, A334499-A334515, A070939, A268754, A363121.

Formula

It seems that a(2*n+2) = A268754(n) and a(2*n+1) = (2*n+1) * 2^A070939(n) = A363121(n+1)/2 for n > 0. - Andrey Zabolotskiy, Sep 04 2024

Extensions

a(11)-a(40) from Jinyuan Wang, May 09 2020
More terms from Bert Dobbelaere, May 09 2020
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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