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A299475 a(n) is the number of vertices in the diagram of partitions of n (see example). 6
1, 4, 7, 10, 16, 22, 34, 46, 67, 91, 127, 169, 232, 304, 406, 529, 694, 892, 1156, 1471, 1882, 2377, 3007, 3766, 4726, 5875, 7309, 9031, 11155, 13696, 16813, 20527, 25048, 30430, 36931, 44650, 53932, 64912, 78046, 93556, 112015, 133750, 159523, 189784, 225526, 267403, 316675, 374263, 441820, 520576, 612679 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

For n >= 1, A299474(n) is the number of edges and A000041(n) is the number of regions in the mentioned diagram (see example and Euler's formula).

LINKS

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

FORMULA

a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.

a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).

EXAMPLE

Construction 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)

a(n)......:   4     7      10        16          22            34     (vertices)

A299474(n):   4     8      12        20          28            44     (edges)

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

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

r     p(n)

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

.             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _

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

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

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

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

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

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

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

8                                                         |_ _|   |   |

9                                                         |_ _ _ _|   |

10                                                        |_ _ _|     |

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

.

Apart from the axis x, the r-th horizontal line segment has length A141285(r), equaling the largest part of the r-th region of the diagram.

Apart from the axis y, the r-th vertical line segment has length A194446(r), equaling the number of parts in the r-th region of the diagram.

The total number of parts equals the sum of largest parts.

Note that every diagram contains all previous diagrams.

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

PROG

(PARI) a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018

CROSSREFS

Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.

Sequence in context: A310711 A115288 A310712 * A310713 A180080 A153003

Adjacent sequences:  A299472 A299473 A299474 * A299476 A299477 A299478

KEYWORD

nonn

AUTHOR

Omar E. Pol, Feb 11 2018

STATUS

approved

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Last modified March 28 14:58 EDT 2020. Contains 333089 sequences. (Running on oeis4.)