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

0, 3, 10, 21, 42, 70, 122, 187, 298, 443, 667, 957, 1401, 1960, 2775, 3828, 5295, 7167, 9745, 12998, 17380, 22915, 30196, 39347, 51274, 66126, 85209, 108942, 139055, 176273, 223148, 280733, 352623, 440646, 549597, 682411, 845852, 1044084, 1286512, 1579582

Offset

0,2

Comments

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = n * A000041(n) + A000217(n) + A006128(n) = A066186(n) + A000217(n) + A006128(n).

Example

For n = 5 the three views of the three-dimensional shell model of partitions with 5 shells look like this:
.
.   A066186(5) = 35     A006128(5) = 20
.
.         1 1 1 1 1     5
.         1 1 1 1 1     3 2
.         1 1 1 1 1     4 1
.         1 1 1 1 1     2 2 1
.         1 1 1 1 1     3 1 1
.         1 1 1 1 1     2 1 1 1
.         1 1 1 1 1     1 1 1 1 1
.
.
.         7 6 4 2 1
.           1 2 3 2
.             1 1 2
.               1 1
.                 1
.
.   A000217(5) = 15
.
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.

Maple

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, n]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= n-> n*b(n, n)[1] +b(n, n)[2] +n*(n+1)/2:
seq (a(n), n=0..50);  # Alois P. Heinz, Mar 22 2012

Mathematica

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 *)

Crossrefs

Other versions: A210970, A210979, A210980, A210990, A210991.

Cf. A000041, A000217, A006128, A026792, A066186, A135010, A138121, A141285, A182703, A182715, A206437, A209655.

Sequence in context: A338089 A294365 A210980 * A268348 A117495 A007687

Adjacent sequences: A207377 A207378 A207379 * A207381 A207382 A207383

Keyword

nonn

Author

Omar E. Pol, Feb 17 2012

Extensions

More terms from Alois P. Heinz, Mar 22 2012

Status

approved