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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 12 12:18 EDT 2020. Contains 335661 sequences. (Running on oeis4.)