%I #29 Nov 04 2013 18:06:27

%S 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,

%T 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,

%U 6,4,5,1,4,1,8,7,4,1,7,2,6,1,5,1,10,42

%N First differences of A225600. Also A141285 and A194446 interleaved.

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

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa408.jpg">Visualization of regions in a minimalist diagram for A006128</a>

%H N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>

%H <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

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

%e 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.

%e 1,1;

%e 2,2;

%e 3,3;

%e 2,1,4,5;

%e 3,1,5,7;

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

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

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

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

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

%e .

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

%e . _ _ _ _ _ _ _

%e . 15 _ _ _ _ |

%e . _ _ _ _|_ |

%e . _ _ _ | |

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

%e . 11 _ _ _ | |

%e . _ _ _|_ | |

%e . _ _ | | |

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

%e . 7 _ _ _ | | |

%e . _ _ _|_ | | |

%e . 5 _ _ | | | |

%e . _ _|_ | | | |

%e . 3 _ _ | | | | |

%e . 2 _ | | | | | |

%e . 1 | | | | | | |

%e .

%e . 1 2 3 4 5 6 7

%e .

%e 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:

%e .

%e 7..................................

%e . /\

%e 5.................... / \ /\

%e . /\ / \ /\ /

%e 3.......... / \ / \ / \/

%e 2..... /\ / \ /\/ \ /

%e 1.. /\ / \ /\/ \ / \ /\/

%e 0 /\/ \/ \/ \/ \/

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

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

%e .

%Y Cf. A000041, A006128, A135010, A138137, A141285, A179862, A186114, A186412, A187219, A194446, A206437, A211978, A220517, A225600, A225610.

%K nonn,tabf

%O 1,3

%A _Omar E. Pol_, Feb 07 2013