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A206440
Volume of the last section of the set of partitions of n from the shell model of partitions version "Boxes".
5
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
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.
Sequence in context: A152535 A042423 A356351 * A192300 A289775 A119503
KEYWORD
nonn,changed
AUTHOR
Omar E. Pol, Feb 08 2012
STATUS
approved