A352285 - OEIS

%I #81 Mar 22 2022 12:32:30

%S 1,1,1,1,3,4,4,4,40,7,58,9,38,8,37,29,71,55,51,41,49,70,60,93,102,79,

%T 333,123,181,69,200,279,372,117,188,212,122,137,263,576,96,149,225,

%U 169,150,276,-1,304,281,106,215,160,206,197,-1,359,221,355,-1,447,178,314,431

%N a(n) is the number of steps in John Conway's game of life that it takes for the smallest square checkerboard pattern with a diagonal of n living cells to either die out or enter a cycle; or -1 if it never cycles.

%C a(n) = -1 iff the pattern's extent grows without bound (since a bounded region must eventually repeat). The first a(n) = -1 is at n=47 where the square launches 8 gliders into open space.

%e For n = 1:

%e . . . | . . . |

%e . o . | . . . |

%e . . . | . . . |

%e all cells are dead after one generation, hence a(1)=1.

%e For n = 2:

%e . . . . | . . . . |

%e . o . . | . . . . |

%e . . o . | . . . . |

%e . . . . | . . . . |

%e all cells are dead after one generation, hence a(2)=1.

%e For n = 3:

%e . . . . .| . . . . . |

%e . o . o .| . . o . . |

%e . . o . .| . o . o . |

%e . o . o .| . . o . . |

%e . . . . .| . . . . . |

%e a pattern repeats after one generation, hence a(3)=1.

%e For n = 4:

%e . . . . . . | . . . . . . |

%e . o . o . . | . . o o . . |

%e . . o . o . | . o . . o . |

%e . o . o . . | . o . . o . |

%e . . o . o . | . . o o . . |

%e . . . . . . | . . . . . . |

%e a pattern repeats after one generation, hence a(4) = 1.

%e For n = 5:

%e . . . . . . . . . | . . . . . . . . . | . . . . . . . . . | . . . . . . . . . |

%e . . . . . . . . . | . . . . . . . . . | . . . . o . . . . | . . . o o o . . . |

%e . . o . o . o . . | . . . o o o . . . | . . . o o o . . . | . . . . . . . . . |

%e . . . o . o . . . | . . o . . . o . . | . . o . o . o . . | . o . . . . . o . |

%e . . o . o . o . . | . . o . . . o . . | . o o o . o o o . | . o . . . . . o . |

%e . . . o . o . . . | . . o . . . o . . | . . o . o . o . . | . o . . . . . o . |

%e . . o . o . o . . | . . . o o o . . . | . . . o o o . . . | . . . . . . . . . |

%e . . . . . . . . . | . . . . . . . . . | . . . . o . . . . | . . . o o o . . . |

%e . . . . . . . . . | . . . . . . . . . | . . . . . . . . . | . . . . . . . . . |

%e a pattern begins to oscillate between four parallel "blinkers" after one generation, hence a(5) = 3.

%Y Cf. A089520 (filled square).

%K sign

%O 1,5

%A _Sebastian F. Orellana_, Mar 10 2022