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URL: https://oeis.org/A299473

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A299473
a(n) = 3*p(n), where p(n) is the number of partitions of n.
4
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
0,1
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).
LINKS
FORMULA
a(n) = 3*A000041(n) = A000041(n) + A139582(n).
a(n) = A299475(n) - 1, n >= 1.
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = 3 * exp(-Pi/24) * 2^(3/8) * Gamma(3/4) / Pi^(1/4) = A389026. - Simon Plouffe, Sep 22 2025
EXAMPLE
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.
CROSSREFS
k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).
Sequence in context: A300300 A293675 A050337 * A355906 A022086 A097135
KEYWORD
nonn
AUTHOR
Omar E. Pol, Feb 10 2018
STATUS
approved