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

1,2

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

Formula

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

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 *)

Crossrefs

Row sums of triangle A206438. Partial sums give A066183.

Cf. A135010, A138121, A141285.

Sequence in context: A152535 A042423 A356351 * A192300 A289775 A119503

Adjacent sequences: A206437 A206438 A206439 * A206441 A206442 A206443

Keyword

nonn

Author

Omar E. Pol, Feb 08 2012

Status

approved