A233968 - OEIS

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A233968

Number of steps between two valleys at height 0 in the infinite Dyck path in which the k-th ascending line segment has A141285(k) steps and the k-th descending line segment has A194446(k) steps, k >= 1.

2, 4, 6, 12, 16, 30, 38, 64, 84, 128, 166, 248, 314, 448, 576, 790, 1004, 1358, 1708, 2264, 2844, 3694, 4614, 5936, 7354, 9342, 11544, 14502, 17816, 22220, 27144, 33584, 40878, 50192, 60828, 74276, 89596, 108778, 130772, 157918, 189116, 227374

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OFFSET

1,1

COMMENTS

Also first differences of A211978.

LINKS

Table of n, a(n) for n=1..42.

FORMULA

a(n) = 2*(A006128(n) - A006128(n-1)) = 2*A138137(n).

EXAMPLE

Illustration of initial terms as a dissection of a minimalist diagram of regions of the set of partitions of n, for n = 1..6:

. _ _ _ _ _ _

. _ _ _ |

. _ _ _|_ |

. _ _ | |

. _ _ _ _ _ | | |

. _ _ _ | |

. _ _ _ _ | | |

. _ _ | | |

. _ _ _ | | | |

. _ _ | | | |

. _ | | | | |

. | | | | | |

. 2 4 6 12 16 30

Also using the elements from the above diagram we can draw an infinite 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 n-th largest peak between two valleys at height 0 is also the partition number A000041(n).

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

. /\

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

. /\ / \ /\ /

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

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

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

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

. 2, 4, 6, 12, 16,...

CROSSREFS

Cf. A000041, A006128, A135010, A138137, A139582, A141285, A182699, A182709, A186412, A194446, A194447, A193870, A206437, A207779, A211009, A211978, A211992, A220517, A225600, A225610, A228109, A228110, A229946.

Sequence in context: A330711 A220219 A284456 * A120453 A326438 A023995

Adjacent sequences: A233965 A233966 A233967 * A233969 A233970 A233971

KEYWORD

nonn

AUTHOR

Omar E. Pol, Jan 14 2014

STATUS

approved