A206440 Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".

1, 5, 11, 27, 43, 93, 131, 247, 352, 584, 808, 1306, 1735, 2643, 3568, 5160, 6835, 9721, 12672, 17564, 22832, 30818, 39743, 53027, 67594, 88740, 112752, 145944, 183979, 236059, 295370, 375208, 467363, 588007, 728437, 910339, 1121009, 1391083, 1706003, 2103013

Offset: 1

Author

Omar E. Pol, Feb 08 2012

Keywords

• nonn

Comments

Since partial sums of this sequence give A066183 we have that A066183(n) is also the volume of the mentioned version of the shell model of partitions with n shells. Each part of size k has a volume equal to k^2 since each box is a cuboid whose sides have lengths: 1, k, k.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Row sums of triangle A206438. Partial sums give A066183.

Cf. A135010, A138121, A141285.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],
      b(n, i-1)+(p-> p+[0, p[1]*i^2])(b(n-i, min(n-i, i))))
    end:
a:= n-> (b(n$2)-b(n-1$2))[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Feb 23 2022

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n},
     b[n, i-1] + Function[p, p + {0, p[[1]]*i^2}][b[n-i, Min[n-i, i]]]];
a[n_] := (b[n, n] - b[n-1, n-1])[[2]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 25 2022, after Alois P. Heinz *)

Formula

a(n) ~ sqrt(3) * zeta(3) * exp(Pi*sqrt(2*n/3)) / Pi^2. - Vaclav Kotesovec, Oct 20 2024