A220517 - OEIS

A220517

First differences of A225600. Also A141285 and A194446 interleaved.

1, 1, 2, 2, 3, 3, 2, 1, 4, 5, 3, 1, 5, 7, 2, 1, 4, 2, 3, 1, 6, 11, 3, 1, 5, 2, 4, 1, 7, 15, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 22, 3, 1, 5, 2, 4, 1, 7, 4, 3, 1, 6, 2, 5, 1, 9, 30, 2, 1, 4, 2, 3, 1, 6, 4, 5, 1, 4, 1, 8, 7, 4, 1, 7, 2, 6, 1, 5, 1, 10, 42

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OFFSET

1,3

COMMENTS

Number of toothpicks added at n-th stage to the toothpick structure (related to integer partitions) of A225600.

LINKS

Table of n, a(n) for n=1..84.

Omar E. Pol, Visualization of regions in a minimalist diagram for A006128

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to toothpick sequences

FORMULA

a(2n-1) = A141285(n); a(2n) = A194446(n), n >= 1

EXAMPLE

Written as an irregular triangle in which row n has length 2*A187219(n) we can see that the right border gives A000041 and the previous term of the last term in row n is n.

1,1;

2,2;

3,3;

2,1,4,5;

3,1,5,7;

2,1,4,2,3,1,6,11;

3,1,5,2,4,1,7,15;

2,1,4,2,3,1,6,4,5,1,4,1,8,22;

3,1,5,2,4,1,7,4,3,1,6,2,5,1,9,30;

2,1,4,2,3,1,6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42;

Illustration of the first seven rows of triangle as a minimalist diagram of regions of the set of partitions of 7:

. _ _ _ _ _ _ _

. 15 _ _ _ _ |

. _ _ _ _|_ |

. _ _ _ | |

. _ _ _|_ _|_ |

. 11 _ _ _ | |

. _ _ _|_ | |

. _ _ | | |

. _ _|_ _|_ | |

. 7 _ _ _ | | |

. _ _ _|_ | | |

. 5 _ _ | | | |

. _ _|_ | | | |

. 3 _ _ | | | | |

. 2 _ | | | | | |

. 1 | | | | | | |

. 1 2 3 4 5 6 7

Also using the elements of this diagram we can draw a Dyck path in which the n-th odd-indexed segment has A141285(n) up-steps and the n-th even-indexed segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two valleys at height 0 is also the partition number A000041(n). See below:

7..................................

. /\

5.................... / \ /\

. /\ / \ /\ /

3.......... / \ / \ / \/

2..... /\ / \ /\/ \ /

1.. /\ / \ /\/ \ / \ /\/

0 /\/ \/ \/ \/ \/

. 0,2, 6, 12, 24, 40... = A211978

. 1, 4, 9, 19, 33... = A179862

CROSSREFS