Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #52 Dec 31 2020 11:11:15
%S 1,2,4,5,7,8,10,11,14,16,18,19,21,23,26,27,29,30,32,33,37,39,41,42,45,
%T 47,51,52,54,55,57,58,62,64,67,68,70,72,76,77,79,80,82,84,87,89,91,92,
%U 95,98,102,104,106,107,111,112,116,118,120,121,123,125,130,131,135,136,138,140,144,147,149,150,152,154
%N a(n) is the total number of regions (or parts) after n-th stage in the diagram of the symmetries of sigma described in A236104.
%C The total area (or total number of cells) of the diagram after n stages is equal to A024916(n), the sum of all divisors of all positive integers <= n.
%C Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(n), see example.
%C For more information see A237593 and A237270.
%C a(n) is also the total number of terraces of the stepped pyramid with n levels described in A245092. - _Omar E. Pol_, Apr 20 2016
%H Robert Price, <a href="/A237590/b237590.txt">Table of n, a(n) for n = 1..5000</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr01.jpg">An infinite stepped pyramid</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr02.jpg">Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)</a>
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the stepped pyramid (first 16 levels)</a>
%F a(n) = A317109(n) - A294723(n) + 1 (Euler's formula). - _Omar E. Pol_, Jul 21 2018
%e Illustration of initial terms:
%e . _ _ _ _
%e . _ _ _ |_ _ _ |_
%e . _ _ _ |_ _ _| |_ _ _| |_
%e . _ _ |_ _ |_ |_ _ |_ _ |_ _ |_ _ |
%e . _ _ |_ _|_ |_ _|_ | |_ _|_ | | |_ _|_ | | |
%e . _ |_ | |_ | | |_ | | | |_ | | | | |_ | | | | |
%e . |_| |_|_| |_|_|_| |_|_|_|_| |_|_|_|_|_| |_|_|_|_|_|_|
%e .
%e .
%e . 1 2 4 5 7 8
%e .
%e For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
%e The sum of all divisors of all positive integers <= 6 is [1] + [1+2] + [1+3] + [1+2+4] + [1+5] + [1+2+3+6] = 33. On the other hand after 6 stages the sum of all parts of the diagram is [1] + [3] + [2+2] + [7] + [3+3] + [12] = 33, equaling the sum of all divisors of all positive integers <= 6.
%e Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
%e From _Omar E. Pol_, Dec 25 2020: (Start)
%e Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
%e ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _|
%e |_ _ _ _ _ _ _ _ _ _ _ _ _ _ |
%e |_ _ _ _ _ _ _ _ _ _ _ _ _ _| |
%e |_ _ _ _ _ _ _ _ _ _ _ _ _ | |
%e |_ _ _ _ _ _ _ _ _ _ _ _ _| | |
%e |_ _ _ _ _ _ _ _ _ _ _ _ | | |_ _ _
%e |_ _ _ _ _ _ _ _ _ _ _ _| | |_ _ _ |
%e |_ _ _ _ _ _ _ _ _ _ _ | | |_ _ | |_
%e |_ _ _ _ _ _ _ _ _ _ _| | |_ _ _| |_ |_
%e |_ _ _ _ _ _ _ _ _ _ | | |_ _| |_
%e |_ _ _ _ _ _ _ _ _ _| | |_ _ |_ |_ _ |_ _
%e |_ _ _ _ _ _ _ _ _ | |_ _ _| |_ | |_ _ |
%e |_ _ _ _ _ _ _ _ _| | |_ _ |_ |_|_ _ | |
%e |_ _ _ _ _ _ _ _ | |_ _ |_ _|_ | | | |_ _ _ _ _ _
%e |_ _ _ _ _ _ _ _| | | | |_ _ | |_|_ _ _ _ _ | |
%e |_ _ _ _ _ _ _ | |_ _ |_ |_ | | |_ _ _ _ _ | | | |
%e |_ _ _ _ _ _ _| |_ _ |_ |_ _ | | |_ _ _ _ _ | | | | | |
%e |_ _ _ _ _ _ | |_ |_ |_ | |_|_ _ _ _ | | | | | | | |
%e |_ _ _ _ _ _| |_ _| |_ | |_ _ _ _ | | | | | | | | | |
%e |_ _ _ _ _ | |_ _ | |_ _ _ _ | | | | | | | | | | | |
%e |_ _ _ _ _| |_ | |_|_ _ _ | | | | | | | | | | | | | |
%e |_ _ _ _ |_ _|_ |_ _ _ | | | | | | | | | | | | | | | |
%e |_ _ _ _| |_ | |_ _ _ | | | | | | | | | | | | | | | | | |
%e |_ _ _ |_ |_|_ _ | | | | | | | | | | | | | | | | | | | |
%e |_ _ _| |_ _ | | | | | | | | | | | | | | | | | | | | | |
%e |_ _ |_ _ | | | | | | | | | | | | | | | | | | | | | | | |
%e |_ _|_ | | | | | | | | | | | | | | | | | | | | | | | | | |
%e |_ | | | | | | | | | | | | | | | | | | | | | | | | | | | |
%e |_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|
%e .
%e (End)
%t (* total number of parts in the first n symmetric representations *)
%t (* Function a237270[] is defined in A237270 *)
%t (* variable "previous" represents the sum from 1 through m-1 *)
%t a237590[previous_,{m_,n_}]:=Rest[FoldList[Plus[#1,Length[a237270[#2]]]&,previous,Range[m,n]]]
%t a237590[n_]:=a237590[0,{1,n}]
%t a237590[78] (* data *)
%t (* _Hartmut F. W. Hoft_, Jul 07 2014 *)
%Y Partial sums of A237271.
%Y Compare with A060831 (analog for the diagram that contains subparts).
%Y Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.
%K nonn
%O 1,2
%A _Omar E. Pol_, Mar 31 2014
%E Definition clarified by _Omar E. Pol_, Jul 21 2018