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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
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
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):
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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)
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r p(n)
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. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
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
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 11 2018
STATUS
approved