A299474 - OEIS

A299474

a(n) = 4*p(n), where p(n) is the number of partitions of n.

4, 4, 8, 12, 20, 28, 44, 60, 88, 120, 168, 224, 308, 404, 540, 704, 924, 1188, 1540, 1960, 2508, 3168, 4008, 5020, 6300, 7832, 9744, 12040, 14872, 18260, 22416, 27368, 33396, 40572, 49240, 59532, 71908, 86548, 104060, 124740, 149352, 178332, 212696, 253044, 300700, 356536, 422232, 499016, 589092, 694100, 816904

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OFFSET

0,1

COMMENTS

For n >= 1, a(n) is also the number of edges in the diagram of partitions of n, in which A299475(n) is the number of vertices and A000041(n) is the number of regions (see example and Euler's formula).

LINKS

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

FORMULA

a(n) = 4*A000041(n) = 2*A139582(n).

a(n) = A000041(n) + A299475(n) - 1, n >= 1 (Euler's formula).

a(n) = A000041(n) + A299473(n). - Omar E. Pol, Aug 11 2018

EXAMPLE

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

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

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

a(n)......: 4 8 12 20 28 44 (edges)

A299475(n): 4 7 10 16 22 34 (vertices)

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.

MAPLE

with(combinat): seq(4*numbpart(n), n=0..50); # Muniru A Asiru, Jul 10 2018

MATHEMATICA

4*PartitionsP[Range[0, 50]] (* Harvey P. Dale, Dec 05 2023 *)

PROG

(GAP) List([0..50], n->4*NrPartitions(n)); # Muniru A Asiru, Jul 10 2018

(PARI) a(n) = 4*numbpart(n); \\ Michel Marcus, Jul 15 2018

(Python)

from sympy.ntheory import npartitions

def a(n): return 4*npartitions(n)

print([a(n) for n in range(51)]) # Michael S. Branicky, Apr 04 2021

CROSSREFS

k times partition numbers: A000041 (k=1), A139582 (k=2), A299473 (k=3), this sequence (k=4).

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

Sequence in context: A194696 A302681 A002368 * A022087 A333149 A095294

Adjacent sequences: A299471 A299472 A299473 * A299475 A299476 A299477

KEYWORD

nonn

AUTHOR

Omar E. Pol, Feb 10 2018

STATUS

approved