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

Offset: 0

Author

Omar E. Pol, Feb 10 2018

Keywords

• nonn

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).

Examples

			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.
		

Links

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

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.

Programs

GAP

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

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 *)

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

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