login
Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.
17

%I #37 Aug 05 2013 03:39:35

%S 1,4,10,18,33,52,87,130,202,295,436,617,887,1226,1709,2327,3173,4244,

%T 5691,7505,9907,12917,16822,21690,27947,35685,45506,57625,72836,91500,

%U 114760,143143,178235,220908,273268,336670,414041,507298,620455,756398,920470

%N Total number of parts in all partitions of n plus the sum of largest parts in all partitions of n plus the number of partitions of n plus n.

%C a(n) is also the total number of toothpicks in a toothpick structure which represents a diagram of regions of the set of partitions of n, n >= 1. The number of horizontal toothpicks is A225596(n). The number of vertical toothpicks is A093694(n). The difference between vertical toothpicks and horizontal toothpicks is A000041(n) - n = A000094(n+1). The total area (or total number of cells) of the diagram is A066186(n). The number of parts in the k-th region is A194446(k). The area (or number of cells) of the k-th region is A186412(k). For the definition of "region" see A206437. For a minimalist version of the diagram (which can be transformed into a Dyck path) see A211978. See also A225600.

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</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(n) = 2*A006128(n) + A000041(n) + n = A211978(n) + A133041(n) = A093694(n) + A006128(n) + n = A093694(n) + A225596(n).

%e For n = 7 the total number of parts in all partitions of 7 plus the sum of largest parts in all partitions of 7 plus the number of partitions of 7 plus 7 is equal to A006128(7) + A006128(7) + A000041(7) + 7 = 54 + 54 + 15 + 7 = 130. On the other hand the number of toothpicks in the diagram of regions of the set of partitions of 7 is equal to 130, so a(7) = 130.

%e . Diagram of regions

%e Partitions of 7 and partitions of 7

%e . _ _ _ _ _ _ _

%e 7 15 |_ _ _ _ |

%e 4 + 3 |_ _ _ _|_ |

%e 5 + 2 |_ _ _ | |

%e 3 + 2 + 2 |_ _ _|_ _|_ |

%e 6 + 1 11 |_ _ _ | |

%e 3 + 3 + 1 |_ _ _|_ | |

%e 4 + 2 + 1 |_ _ | | |

%e 2 + 2 + 2 + 1 |_ _|_ _|_ | |

%e 5 + 1 + 1 7 |_ _ _ | | |

%e 3 + 2 + 1 + 1 |_ _ _|_ | | |

%e 4 + 1 + 1 + 1 5 |_ _ | | | |

%e 2 + 2 + 1 + 1 + 1 |_ _|_ | | | |

%e 3 + 1 + 1 + 1 + 1 3 |_ _ | | | | |

%e 2 + 1 + 1 + 1 + 1 + 1 2 |_ | | | | | |

%e 1 + 1 + 1 + 1 + 1 + 1 + 1 1 |_|_|_|_|_|_|_|

%e .

%e . 1 2 3 4 5 6 7

%e .

%e Illustration of initial terms as the number of toothpicks in a diagram of regions of the set of partitions of n, for n = 1..6:

%e . _ _ _ _ _ _

%e . |_ _ _ |

%e . |_ _ _|_ |

%e . |_ _ | |

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

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

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

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

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

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

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

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

%e .

%e . 4 10 18 33 52 87

%Y Cf. A000041, A000094, A006128, A066186, A093694, A133041, A135010, A138137, A139250, A139582, A141285, A182377, A186114, A186412, A187219, A194446, A194447, A206437, A207779, A211978, A220517, A225596, A225600.

%K nonn

%O 0,2

%A _Omar E. Pol_, Jul 29 2013