A330155 - OEIS

%I #31 Jan 06 2020 03:46:38

%S 1,1,2,2,3,1,2,1,3,4,3,1,5,2,4,3,2,6,5,1,4,4,2,1,6,5,7,3,4,3,1,8,6,5,

%T 7,2,5,3,2,9,8,6,4,7,1,5,4,2,1,9,8,6,3,7,10,6,4,3,1,11,9,8,5,2,7,10,6,

%U 5,3,2,12,11,9,8,4,1,7,10,7,5,4,2,1,12,11,9,8,3,13,6,10

%N Triangle read by rows. Given n enumerated cards in a stack, with 1 at the top and n at the bottom, repeat the following process k times: remove the card in the middle (at position (size of the stack)/2, rounding up), and move the card at the bottom of the stack to the top. T(n,k) is the number of the last card removed.

%F T(n,n) = A308432(n), n > 0.

%F Conjecture: (Start)

%F Each diagonal forms a unique sequence S. Let S(m) be the m-th diagonal in T, for example with m=2, S(2) = 1,3,3,2,1,7,...; then T(n,k) = k-th element in S(n-k+1).

%F Let z = ceiling(m/2); the first z elements in S(m) are z,z-1,z-2,...,1.

%F Let G(x) = 3*((x-2)/2)+2 if x even,

%F            3*((x-1)/2)+1 otherwise.

%F Let B(x) = Sum_{i=0..x-1} 2*G(m)*3^i.

%F Let C(x) = z if x=0,

%F            B(x)+z otherwise.

%F C(x)-th element in S(m) is 1, for all x >= 0.

%F Let D(x) = G(m)*3^(x-1), with x > 0.

%F Let y = minimum x such that k <= C(x).

%F Finally S(m) = z-k+1 if z >= k,

%F                D(y)+1 if C(y)-k >= D(y),

%F                C(y)-k+1 otherwise.

%F         for all k.

%F Then T(n,k) = k-th element in S(n-k+1).

%F (End)

%e Triangle begins:

%e   1;

%e   1,  2;

%e   2,  3,  1;

%e   2,  1,  3,  4;

%e   3,  1,  5,  2,  4;

%e   3,  2,  6,  5,  1,  4;

%e   4,  2,  1,  6,  5,  7,  3;

%e   4,  3,  1,  8,  6,  5,  7,  2;

%e   5,  3,  2,  9,  8,  6,  4,  7,  1;

%e   5,  4,  2,  1,  9,  8,  6,  3,  7, 10;

%e   ...

%e With n=5, row #5 is 3,1,5,2,4. In the diagram below, each "X" represents the removal of a card:

%e .

%e                           +-->4X

%e                           |

%e                    +-->2X |

%e                    |      |

%e             +-->4--+-->4--+

%e             |      |

%e      +-->5--+-->5X |

%e      |      |      |

%e   1--+-->1X |      |

%e      |      |      |

%e   2--+-->2--+-->2--+

%e      |      |

%e   3X |      |

%e      |      |

%e   4--+-->4--+

%e      |

%e   5--+

%Y This triangle is based on A308432.

%K nonn,tabl

%O 1,3

%A _Wilmer Emiro Castrillon Calderon_, Dec 03 2019