A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

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

Offset: 0

Author

Omar E. Pol, Feb 11 2018

Keywords

• nonn

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).

Examples

			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.
		

Links

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

Crossrefs

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

Programs

PARI

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

Formula

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

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