This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A182738 #56 Oct 02 2023 20:16:01
%S A182738 1,5,14,34,69,135,240,416,686,1106,1722,2646,3959,5849,8489,12185,
%T A182738 17234,24164,33474,46014,62646,84690,113555,151355,200305,263641,
%U A182738 344911,449015,581400,749520,961622,1228790,1563509,1982049
%N A182738 Partial sums of A066186.
%C A182738 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.
%C A182738 From _Omar E. Pol_, Jan 20 2021: (Start)
%C A182738 a(n) is the sum of all parts of all partitions of all positive integers <= n.
%C A182738 Convolution of A000203 and A000070.
%C A182738 Convolution of A024916 and A000041.
%C A182738 Convolution of A175254 and A002865.
%C A182738 Convolution of A340793 and A014153.
%C A182738 Row sums of triangles A340527, A340531, A340579.
%C A182738 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.
%C A182738 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).
%C A182738 a(n) is the volume (or the total number of unit cubes) of the polycube.
%C A182738 That is due to the correspondence between divisors and partitions (cf. A336811).
%C A182738 The symmetric tower is a member of the family of the pyramid described in A245092.
%C A182738 The growth of the volume of the polycube represents every convolution mentioned above. (End)
%H A182738 Vaclav Kotesovec, <a href="/A182738/b182738.txt">Table of n, a(n) for n = 1..10000</a>
%H A182738 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpatru.jpg">Illustration of a(6) = 135</a>, the polycube contains 135 unit cubes.
%H A182738 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of a(9) = 686</a>, the polycube contains 686 unit cubes.
%F A182738 a(n) = n*A000070(n) - A014153(n-1). - _Vaclav Kotesovec_, Jun 23 2015
%F A182738 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
%F A182738 G.f.: x*f'(x)/(1 - x), where f(x) = Product_{k>=1} 1/(1 - x^k). - _Ilya Gutkovskiy_, Apr 10 2017
%e A182738 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).
%t A182738 With[{no=35},Accumulate[PartitionsP[Range[no]]Range[no]]] (* _Harvey P. Dale_, Feb 02 2011 *)
%Y A182738 Cf. A000041, A000070, A000203, A002865, A014153, A024916, A066186, A135010, A175254, A237270, A237593, A245092, A336811, A340527, A340531, A340579, A340793.
%K A182738 nonn,easy
%O A182738 1,2
%A A182738 _Omar E. Pol_, Jan 22 2011