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A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.

%I A207380 #59 Jun 18 2015 08:30:14
%S A207380 0,3,10,21,42,70,122,187,298,443,667,957,1401,1960,2775,3828,5295,
%T A207380 7167,9745,12998,17380,22915,30196,39347,51274,66126,85209,108942,
%U A207380 139055,176273,223148,280733,352623,440646,549597,682411,845852,1044084,1286512,1579582
%N A207380 Total area of the shadows of the three views of a three-dimensional version of the shell model of partitions with n shells.
%C A207380 In this model each part of a partition can be represented by a cuboid of size 1 x 1 x L, where L is the size of the part. One of the views is a rectangle formed by ones whose area is n*A000041(n) = A066186(n). Each element of the first view is equal to the volume of a horizontal column parallel to the axis x. The second view is the n-th slice illustrated in A026792 which has A000041(n) levels and its area is A006128(n) equals the total number of parts of all partitions of n and equals the sum of largest parts of all partitions of n. Each zone contains a partition of n. Each element of the second view is equal to the volume of a horizontal column parallel to the axis y. The third view is a triangle because it is also the n-th slice of the tetrahedron of A209655. The area of triangle is A000217(n). Each element of the third view is equal to the volume of a vertical column parallel to the axis z. The sum of elements of each view is A066186(n) equals the area of the first view. For more information about the shell model of partitions see A135010 and A182703.
%H A207380 Alois P. Heinz, <a href="/A207380/b207380.txt">Table of n, a(n) for n = 0..1000</a>
%F A207380 a(n) = n * A000041(n) + A000217(n) + A006128(n) = A066186(n) + A000217(n) + A006128(n).
%e A207380 For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
%e A207380 .
%e A207380 .   A066186(5) = 35     A006128(5) = 20
%e A207380 .
%e A207380 .         1 1 1 1 1     5
%e A207380 .         1 1 1 1 1     3 2
%e A207380 .         1 1 1 1 1     4 1
%e A207380 .         1 1 1 1 1     2 2 1
%e A207380 .         1 1 1 1 1     3 1 1
%e A207380 .         1 1 1 1 1     2 1 1 1
%e A207380 .         1 1 1 1 1     1 1 1 1 1
%e A207380 .
%e A207380 .
%e A207380 .         7 6 4 2 1
%e A207380 .           1 2 3 2
%e A207380 .             1 1 2
%e A207380 .               1 1
%e A207380 .                 1
%e A207380 .
%e A207380 .   A000217(5) = 15
%e A207380 .
%e A207380 The areas of the shadows of the three views are A066186(5) = 35, A006128(5) = 20 and A000217(5) = 15, therefore the total area of the three shadows is 35+20+15 = 70, so a(5) = 70.
%p A207380 b:= proc(n, i) option remember; local f, g;
%p A207380       if n=0 or i=1 then [1, n]
%p A207380     else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
%p A207380          [f[1]+g[1], f[2]+g[2]+g[1]]
%p A207380       fi
%p A207380     end:
%p A207380 a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
%p A207380 seq (a(n), n=0..50);  # _Alois P. Heinz_, Mar 22 2012
%t A207380 b[n_, i_] := b[n, i] = Module[{f, g}, If [n == 0 || i == 1, {1, n}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; Join[f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]] ]]]; a[n_] := n*b[n, n][[1]] + b[n, n][[2]] + n*(n+1)/2; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Jun 18 2015, after _Alois P. Heinz_ *)
%Y A207380 Other versions: A210970, A210979, A210980, A210990, A210991.
%Y A207380 Cf. A000041, A000217, A006128, A026792, A066186, A135010, A138121, A141285, A182703, A182715, A206437, A209655.
%K A207380 nonn
%O A207380 0,2
%A A207380 _Omar E. Pol_, Feb 17 2012
%E A207380 More terms from _Alois P. Heinz_, Mar 22 2012