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

%I A299473 #34 Jul 15 2018 13:38:52
%S A299473 3,3,6,9,15,21,33,45,66,90,126,168,231,303,405,528,693,891,1155,1470,
%T A299473 1881,2376,3006,3765,4725,5874,7308,9030,11154,13695,16812,20526,
%U A299473 25047,30429,36930,44649,53931,64911,78045,93555,112014,133749,159522,189783,225525,267402,316674,374262,441819,520575,612678
%N A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n.
%C A299473 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).
%H A299473 Shawn A. Broyles, <a href="/A299473/b299473.txt">Table of n, a(n) for n = 0..1000</a>
%F A299473 a(n) = 3*A000041(n) = A000041(n) + A139582(n).
%F A299473 a(n) = A299475(n) - 1, n >= 1.
%e A299473 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):
%e A299473 -----------------------------------------------------------------------------------
%e A299473 n.........:    1     2       3         4           5           6   (stage)
%e A299473 A000041(n):    1     2       3         5           7          11   (open regions)
%e A299473 A139582(n):    2     4       6        10          14          22   (line segments)
%e A299473 a(n)......:    3     6       9        15          21          33   (vertices)
%e A299473 -----------------------------------------------------------------------------------
%e A299473 r     p(n)
%e A299473 -----------------------------------------------------------------------------------
%e A299473 .
%e A299473 1 .... 1 .... _|   _| |   _| | |   _| | | |   _| | | | |   _| | | | | |
%e A299473 2 .... 2 ......... _ _|   _ _| |   _ _| | |   _ _| | | |   _ _| | | | |
%e A299473 3 .... 3 ................ _ _ _|   _ _ _| |   _ _ _| | |   _ _ _| | | |
%e A299473 4                                  _ _|   |   _ _|   | |   _ _|   | | |
%e A299473 5 .... 5 ......................... _ _ _ _|   _ _ _ _| |   _ _ _ _| | |
%e A299473 6                                             _ _ _|   |   _ _ _|   | |
%e A299473 7 .... 7 .................................... _ _ _ _ _|   _ _ _ _ _| |
%e A299473 8                                                          _ _|   |   |
%e A299473 9                                                          _ _ _ _|   |
%e A299473 10                                                         _ _ _|     |
%e A299473 11 .. 11 ................................................. _ _ _ _ _ _|
%e A299473 .
%e A299473 The r-th horizontal line segment has length A141285(r).
%e A299473 The r-th vertical line segment has length A194446(r).
%e A299473 An infinite diagram is a minimalist table of all partitions of all positive integers.
%Y A299473 k times partition numbers: A000041 (k=1), A139582 (k=2), this sequence (k=3), A299474 (k=4).
%Y A299473 Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475.
%K A299473 nonn
%O A299473 0,1
%A A299473 _Omar E. Pol_, Feb 10 2018