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