A206440 - OEIS

%I #32 Oct 20 2024 09:21:18

%S 1,5,11,27,43,93,131,247,352,584,808,1306,1735,2643,3568,5160,6835,

%T 9721,12672,17564,22832,30818,39743,53027,67594,88740,112752,145944,

%U 183979,236059,295370,375208,467363,588007,728437,910339,1121009,1391083,1706003,2103013

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

%C 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.

%H Alois P. Heinz, <a href="/A206440/b206440.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) ~ sqrt(3) * zeta(3) * exp(Pi*sqrt(2*n/3)) / Pi^2. - _Vaclav Kotesovec_, Oct 20 2024

%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n],

%p       b(n, i-1)+(p-> p+[0, p[1]*i^2])(b(n-i, min(n-i, i))))

%p     end:

%p a:= n-> (b(n$2)-b(n-1$2))[2]:

%p seq(a(n), n=1..40);  # _Alois P. Heinz_, Feb 23 2022

%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n},

%t      b[n, i-1] + Function[p, p + {0, p[[1]]*i^2}][b[n-i, Min[n-i, i]]]];

%t a[n_] := (b[n, n] - b[n-1, n-1])[[2]];

%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Apr 25 2022, after _Alois P. Heinz_ *)

%Y Row sums of triangle A206438. Partial sums give A066183.

%Y Cf. A135010, A138121, A141285.

%K nonn

%O 1,2

%A _Omar E. Pol_, Feb 08 2012