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%I #15 May 23 2012 17:36:58
%S 0,3,10,21,42,69,123,189,304,458,693,998,1474,2067,2927,4056,5613,
%T 7595,10335,13782,18411,24276,31944,41583,54152,69762,89758,114668,
%U 146181,185083,234051,294126,368992,460669,573906,711865,881506,1087023,1338043
%N Total area of the shadows of the three views of the shell model of partitions, version "Tree", with n shells.
%C Each part is represented by a cuboid 1 X 1 X L where L is the size of the part.
%F a(n) = A066186(n) + A194804(n) + A194805(n), n >= 1.
%e For n = 7 the shadows of the three views of the shell model of partitions version "tree" with seven shells looks like this:
%e . | Partitions
%e . A194805(7) = 25 A066186(7) = 105 | of 7
%e . |
%e . 1 * * * * * * 1 | 7
%e . 2 * * * 1 * * 2 | 4+3
%e . 2 * * * * 1 * 2 | 5+2
%e . 3 * * 1 * 2 * 3 | 3+2+2
%e . 1 2 * * * * * 1 2 | 6+1
%e . 2 3 * * 1 * * 2 3 | 3+3+1
%e . 2 3 * * * 1 * 2 3 | 4+2+1
%e . 3 4 * 1 * 2 * 3 4 | 2+2+2+1
%e . 3 1 * * * * 1 2 3 | 5+1+1
%e . 4 2 * * 1 * 2 3 4 | 3+2+1+1
%e . 1 4 * * * 1 2 3 4 | 4+1+1+1
%e . 2 5 * 1 * 2 3 4 5 | 2+2+1+1+1
%e . 5 1 * * 1 2 3 4 5 | 3+1+1+1+1
%e . 1 6 * 1 2 3 4 5 6 | 2+1+1+1+1+1
%e . 7 1 2 3 4 5 6 7 | 1+1+1+1+1+1+1
%e . ---------------------------------- |
%e . |
%e . * * * * 1 * * * * |
%e . * * * 1 2 * * * * |
%e . * 1 * * 2 1 * * * |
%e . * * 1 2 2 * * 1 * |
%e . * * * * 2 2 1 * * |
%e . 1 2 2 3 2 * * * * |
%e . 2 3 2 2 1 |
%e . |
%e . A194804(7) = 59 |
%e .
%e Note that, as a variant, in this case each part is labeled with its position in the partition.
%e The areas of the shadows of the three views are A066186(7) = 105, A194804(7) = 59 and A194805(7) = 25, therefore the total area of the three shadows is 105+59+25 = 189, so a(7) = 189.
%Y Other versions: A207380, A210970, A210979, A210990, A210991.
%Y Cf. A000041, A066186, A135010, A138121, A141285, A182703, A194804, A194805, A206437.
%K nonn
%O 0,2
%A _Omar E. Pol_, Apr 21 2012
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