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%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
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