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A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).

%I A299475 #35 Jul 15 2018 13:27:12
%S A299475 1,4,7,10,16,22,34,46,67,91,127,169,232,304,406,529,694,892,1156,1471,
%T A299475 1882,2377,3007,3766,4726,5875,7309,9031,11155,13696,16813,20527,
%U A299475 25048,30430,36931,44650,53932,64912,78046,93556,112015,133750,159523,189784,225526,267403,316675,374263,441820,520576,612679
%N A299475 a(n) is the number of vertices in the diagram of partitions of n (see example).
%C A299475 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).
%H A299475 Shawn A. Broyles, <a href="/A299475/b299475.txt">Table of n, a(n) for n = 0..1000</a>
%F A299475 a(0) = 1; a(n) = 1 + 3*A000041(n), n >= 1.
%F A299475 a(n) = A299474(n) - A000041(n) + 1, n >= 1 (Euler's formula).
%e A299475 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):
%e A299475 --------------------------------------------------------------------------------
%e A299475 n ........:   1     2       3         4           5             6     (stage)
%e A299475 a(n)......:   4     7      10        16          22            34     (vertices)
%e A299475 A299474(n):   4     8      12        20          28            44     (edges)
%e A299475 A000041(n):   1     2       3         5           7            11     (regions)
%e A299475 --------------------------------------------------------------------------------
%e A299475 r     p(n)
%e A299475 --------------------------------------------------------------------------------
%e A299475 .             _    _ _    _ _ _    _ _ _ _    _ _ _ _ _    _ _ _ _ _ _
%e A299475 1 .... 1 ....|_|  |_| |  |_| | |  |_| | | |  |_| | | | |  |_| | | | | |
%e A299475 2 .... 2 .........|_ _|  |_ _| |  |_ _| | |  |_ _| | | |  |_ _| | | | |
%e A299475 3 .... 3 ................|_ _ _|  |_ _ _| |  |_ _ _| | |  |_ _ _| | | |
%e A299475 4                                 |_ _|   |  |_ _|   | |  |_ _|   | | |
%e A299475 5 .... 5 .........................|_ _ _ _|  |_ _ _ _| |  |_ _ _ _| | |
%e A299475 6                                            |_ _ _|   |  |_ _ _|   | |
%e A299475 7 .... 7 ....................................|_ _ _ _ _|  |_ _ _ _ _| |
%e A299475 8                                                         |_ _|   |   |
%e A299475 9                                                         |_ _ _ _|   |
%e A299475 10                                                        |_ _ _|     |
%e A299475 11 .. 11 .................................................|_ _ _ _ _ _|
%e A299475 .
%e A299475 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.
%e A299475 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.
%e A299475 The total number of parts equals the sum of largest parts.
%e A299475 Note that every diagram contains all previous diagrams.
%e A299475 An infinite diagram is a table of all partitions of all positive integers.
%o A299475 (PARI) a(n) = if (n==0, 1, 1+3*numbpart(n)); \\ _Michel Marcus_, Jul 15 2018
%Y A299475 Cf. A000041, A135010, A139582, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299474.
%K A299475 nonn
%O A299475 0,2
%A A299475 _Omar E. Pol_, Feb 11 2018