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 A299473 a(n) = 3*p(n), where p(n) is the number of partitions of n. 2
 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 (list; graph; refs; listen; history; text; internal format)
 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 Shawn A. Broyles, Table of n, a(n) for n = 0..1000 FORMULA a(n) = 3*A000041(n) = A000041(n) + A139582(n). a(n) = A299475(n) - 1, n >= 1. 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). Cf. A135010, A141285, A182181, A186114, A193870, A194446, A194447, A206437, A207779, A220482, A220517, A273140, A278355, A278602, A299475. Sequence in context: A300300 A293675 A050337 * A022086 A097135 A293677 Adjacent sequences:  A299470 A299471 A299472 * A299474 A299475 A299476 KEYWORD nonn AUTHOR Omar E. Pol, Feb 10 2018 STATUS approved

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Last modified April 3 14:34 EDT 2020. Contains 333197 sequences. (Running on oeis4.)