A182738 Partial sums of A066186.

1, 5, 14, 34, 69, 135, 240, 416, 686, 1106, 1722, 2646, 3959, 5849, 8489, 12185, 17234, 24164, 33474, 46014, 62646, 84690, 113555, 151355, 200305, 263641, 344911, 449015, 581400, 749520, 961622, 1228790, 1563509, 1982049

Offset: 1

Author

Omar E. Pol, Jan 22 2011

Keywords

• nonn
• easy

Comments

a(n) is also the volume of a three-dimensional version of the section model of partitions: the 3D illustrations in A135010 show boxes with face areas of 1 X 1, 2 X 2, 3 X 3, 4 X 5, 5 X 7 units along the m and p(m) axis, which is sequence A066186. Assuming that the boxes are 1 unit deep, the total volume of all boxes up to layer n is a(n). See the first two links.

From Omar E. Pol, Jan 20 2021: (Start)

a(n) is the sum of all parts of all partitions of all positive integers <= n.

Convolution of A000203 and A000070.

Convolution of A024916 and A000041.

Convolution of A175254 and A002865.

Convolution of A340793 and A014153.

Row sums of triangles A340527, A340531, A340579.

Consider a symmetric tower (a polycube) in which the terraces are the symmetric representation of sigma (n..1) respectively starting from the base (cf. A237270, A237593). The total area of the terraces equals A024916(n), the same as the area of the base.

The levels of the terraces starting from the base are the first n terms of A000070, that is A000070(0)..A000070(n-1), hence the differences between two successive levels give the partition numbers A000041, that is A000041(0)..A000041(n-1).

a(n) is the volume (or the total number of unit cubes) of the polycube.

That is due to the correspondence between divisors and partitions (cf. A336811).

The symmetric tower is a member of the family of the pyramid described in A245092.

The growth of the volume of the polycube represents every convolution mentioned above. (End)

Examples

			a(6) = 135 because the volume V(6) = p(1) + 2*p(2) + 3*p(3) + 4*p(4) + 5*p(5) + 6*p(6) = 1 + 2*2 + 3*3 + 4*5 + 5*7 + 6*11 = 1 + 4 + 9 + 20 + 35 + 66 = 135 where p(n) = A000041(n).
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
• Omar E. Pol, Illustration of a(6) = 135, the polycube contains 135 unit cubes.
• Omar E. Pol, Illustration of a(9) = 686, the polycube contains 686 unit cubes.

Crossrefs

Cf. A000041, A000070, A000203, A002865, A014153, A024916, A066186, A135010, A175254, A237270, A237593, A245092, A336811, A340527, A340531, A340579, A340793.

Programs

Mathematica

With[{no=35},Accumulate[PartitionsP[Range[no]]Range[no]]] (* Harvey P. Dale, Feb 02 2011 *)

Formula

a(n) = n*A000070(n) - A014153(n-1). - Vaclav Kotesovec, Jun 23 2015

a(n) ~ sqrt(n) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)) * (1 + (11*Pi/(24*sqrt(6)) - sqrt(6)/Pi)/sqrt(n) + (73*Pi^2/6912 - 3/16)/n). - Vaclav Kotesovec, Jun 23 2015, extended Nov 04 2016

G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} 1/(1 - x^k). - Ilya Gutkovskiy, Apr 10 2017