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A237590
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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.
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20
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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, 47, 51, 52, 54, 55, 57, 58, 62, 64, 67, 68, 70, 72, 76, 77, 79, 80, 82, 84, 87, 89, 91, 92, 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
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OFFSET
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1,2
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COMMENTS
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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.
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.
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
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LINKS
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FORMULA
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EXAMPLE
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Illustration of initial terms:
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. 1 2 4 5 7 8
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For n = 6 the diagram contains 8 regions (or parts), so a(6) = 8.
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.
Note that the region between the virtual circumscribed square and the diagram is a symmetric polygon whose area is equal to A004125(6) = 3.
Illustration of the diagram after 29 stages (contain 215 vertices, 268 edges and 54 regions or parts):
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(End)
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MATHEMATICA
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(* total number of parts in the first n symmetric representations *)
(* Function a237270[] is defined in A237270 *)
(* variable "previous" represents the sum from 1 through m-1 *)
a237590[previous_, {m_, n_}]:=Rest[FoldList[Plus[#1, Length[a237270[#2]]]&, previous, Range[m, n]]]
a237590[n_]:=a237590[0, {1, n}]
a237590[78] (* data *)
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CROSSREFS
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Compare with A060831 (analog for the diagram that contains subparts).
Cf. A000203, A004125, A024916, A196020, A236104, A235791, A237048, A237270, A237591, A237593, A239659, A239660, A239663, A239665, A239931-A239934, A245092, A244050, A244970, A262626, A317109.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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