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A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n. 2
3, 3, 6, 9, 15, 21, 33, 45, 66, 90, 126, 168, 231, 303, 405, 528, 693, 891, 1155, 1470, 1881, 2376, 3006, 3765, 4725, 5874, 7308, 9030, 11154, 13695, 16812, 20526, 25047, 30429, 36930, 44649, 53931, 64911, 78045, 93555, 112014, 133749, 159522, 189783, 225525, 267402, 316674, 374262, 441819, 520575, 612678 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

For n >= 1, a(n) is also the number of vertices in the minimalist diagram of partitions of n, in which A139582(n) is the number of line segments and A000041(n) is the number of open regions (see example).

LINKS

Shawn A. Broyles, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = 3*A000041(n) = A000041(n) + A139582(n).

a(n) = A299475(n) - 1, n >= 1.

EXAMPLE

Construction of a minimalist version of a modular table of partitions in which a(n) is the number of vertices of the diagram after n-th stage (n = 1..6):

-----------------------------------------------------------------------------------

n.........:    1     2       3         4           5           6   (stage)

A000041(n):    1     2       3         5           7          11   (open regions)

A139582(n):    2     4       6        10          14          22   (line segments)

a(n)......:    3     6       9        15          21          33   (vertices)

-----------------------------------------------------------------------------------

r     p(n)

-----------------------------------------------------------------------------------

.

1 .... 1 .... _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |

2 .... 2 ......... _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |

3 .... 3 ................ _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |

4                                  _ _|   |   _ _|   | |   _ _|   | | |

5 .... 5 ......................... _ _ _ _|   _ _ _ _| |   _ _ _ _| | |

6                                             _ _ _|   |   _ _ _|   | |

7 .... 7 .................................... _ _ _ _ _|   _ _ _ _ _| |

8                                                          _ _|   |   |

9                                                          _ _ _ _|   |

10                                                         _ _ _|     |

11 .. 11 ................................................. _ _ _ _ _ _|

.

The r-th horizontal line segment has length A141285(r).

The r-th vertical line segment has length A194446(r).

An infinite diagram is a minimalist table of all partitions of all positive integers.

CROSSREFS

k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).

Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.

Sequence in context: A300300 A293675 A050337 * A022086 A097135 A293677

Adjacent sequences:  A299470 A299471 A299472 * A299474 A299475 A299476

KEYWORD

nonn

AUTHOR

Omar E. Pol, Feb 10 2018

STATUS

approved

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Last modified April 3 14:34 EDT 2020. Contains 333197 sequences. (Running on oeis4.)