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%I #21 May 24 2012 13:07:10
%S 0,3,9,18,21,35,39,58,61,67,71,99,103,110,115,152,155,161,165,175,181,
%T 186,238,242,249,254,265,269,277,283,352,355,361,365,375,381,386,401,
%U 406,415,422,428,522,526,533,538,549,553,561,567,584,590,595,606
%N Total area of the shadows of the three views of the shell model of partitions with n regions.
%C It appears that if n is a partition number A000041 then the rotated structure with n regions shows each row as a partition of k such that A000041(k) = n (see example).
%C For the definition of "regions of n" see A206437.
%F a(n) = A182181(n) + A182727(n) + A210692(n).
%F a(A000041(n)) = 2*A006128(n) + A026905(n).
%e For n = 11 the three views of the shell model of partitions with 11 regions look like this:
%e .
%e . A182181(11) = 35 A210692(11) = 29
%e .
%e . 1 1
%e . 1 1
%e . 1 1
%e . 1 1
%e . 1 1 1 1
%e . 1 1 1 1
%e . 1 1 1 1 1 1
%e . 2 1 1 1 1 2
%e . 2 1 1 1 1 1 1 2
%e . 3 2 2 2 1 1 1 1 2 2 3
%e . 6 3 4 2 5 3 4 2 3 2 1 1 2 3 4 5 6
%e . <------- Regions ------ ------------> N
%e . L
%e . a 1
%e . r * 2
%e . g * * 3
%e . e * 2
%e . s * * * 4
%e . t * * 3
%e . * * * * 5
%e . p * 2
%e . a * * * 4
%e . r * * 3
%e . t * * * * * 6
%e . s
%e .
%e . A182727(11) = 35
%e .
%e The areas of the shadows of the three views are A182181(11) = 35, A182727(11) = 35 and A210692(11) = 29, therefore the total area of the three shadows is 35+35+29 = 99, so a(11) = 99.
%e Since n = 11 is a partition number A000041 we can see that the rotated structure with 11 regions shows each row as a partition of 6 because A000041(6) = 11. See below:
%e .
%e . 6
%e . 3 3
%e . 4 2
%e . 2 2 2
%e . 5 1
%e . 3 2 1
%e . 4 1 1
%e . 2 2 1 1
%e . 3 1 1 1
%e . 2 1 1 1 1
%e . 1 1 1 1 1 1
%e .
%Y Other versions: A207380, A210970, A210979, A210980, A210990.
%Y Cf. A000041, A026905, A135010, A138121, A141285, A182703, A194446, A182181, A182727, A186114, A206437, A210692.
%K nonn
%O 0,2
%A _Omar E. Pol_, Apr 30 2012