A331859 The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall.

3, 5, 5, 6, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25

Offset: 1

Author

Peter Kagey, Jan 29 2020

Keywords

• nonn

Comments

Suppose there is a block A of mass n sliding left toward a stationary block B of mass 1, to the left of which is a wall. Assuming the sliding is frictionless and the collisions are elastic, a(n) is the number of collisions between A and B plus the number of collisions between B and the wall. (See Grant Sanderson links for animated examples.)

a(100^n) = A011545(n).

Since arctan(sqrt(1/n)) is approximately sqrt(1/n) for large values of n, a(n) = A121854(n) for most values of n.

Conjecture: The values of n for which a(n) != A121854(n) is a subset of A331903.

Initial phase:

\ | ____________________

\ \| | |

\ | | |

\ \| | |

\ | | |

\ \| <=== | Block A |

\ | _______ | |

\ \| | | | M = n |

\ | | Block B | | |

\ \| | | | | |

\ | | M = 1 | | |

\ \| |_______| |____________________|

\ L----------------------------------------------------------

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\ \|

\ | ____________________

\ \| | |

\ | | |

\ \| | |

\ | | |

\ \| <=== | |

\ | _______ | |

\ \| | || |

\ | | || |

\ \| | || |

\ | | || |

\ \| |_______||____________________|

\ L----------------------------------------------------------

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

\ \|

\ | ____________________

\ \| | |

\ | | |

\ \| | |

\ | | |

\ \| <== | |

\ | _______ | |

\ \| | | | |

\ | | | | |

\ \|<===>| | | |

\ | | | | |

\ \| |_______| |____________________|

\ L----------------------------------------------------------

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Code Golf Stack Exchange, Elastic collisions between blocks
• Grant Sanderson, How Pi Connects Colliding Blocks to a Quantum Search Algorithm, Quanta Magazine (2020).
• Grant Sanderson, The most unexpected answer to a counting puzzle, 3Blue1Brown video (2019)
• Grant Sanderson, Why do colliding blocks compute pi?, 3Blue1Brown video (2019)

Crossrefs

Cf. A121854, A331903, A331904.

Programs

Mathematica

Table[Ceiling[Pi/ArcTan[Sqrt[1/n]] - 1], {n, 1, 100}]

Formula

a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1.