A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.

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

Offset: 0

Author

Omar E. Pol, Feb 10 2018

Keywords

• nonn

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).

Examples

			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.
		

Links

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

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.

Formula

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

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