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A299475
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a(n) is the number of vertices in the diagram of partitions of n (see example).
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6
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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
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OFFSET
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0,2
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COMMENTS
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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).
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LINKS
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FORMULA
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a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.
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EXAMPLE
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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)
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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 .................................................|_ _ _ _ _ _|
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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.
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PROG
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(PARI) a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ Michel Marcus, Jul 15 2018
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CROSSREFS
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Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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