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Search: id:A129757
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| A129757 |
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Maximum genus of fixed edge 2^m-1 binary state graph with 2*m+1 states: Vertices(n)=Floor[2^(n/2)]; Faces(n)=Floor[2^[m-n/2]; Edges(n)=Vertices(n)+Faces(n)-2+2*g=2^m-1; solved for g at the central point m. |
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1
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| 1, 1, 3, 5, 12, 25, 54, 113, 235, 481, 980, 1985, 4007, 8065, 16204, 32513, 65175, 130561, 261421, 523265, 1047129, 2095105, 4191409, 8384513, 16771425, 33546241, 67097280, 134201345, 268412287, 536838145, 1073695485, 2147418113
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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The idea was to get a binary graph system of vertices, edges, and faces that had a genus near the exceptional group sequence dimension. It is a form of combinatorial optimization. The object was to get an idea of what higher dimenional exceptional group dimensions would look like if they existed.
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FORMULA
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a(n) =Ceiling[Flatten[Table[N[Flatten[g /. Solve[2*Floor[2^(n/2)] - 2 + 2*g - (2^n - 1) == 0, g]]], {n, 1, 32}]]]
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EXAMPLE
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Exceptal group dimension to output:
14->12->G2
24 ->25->A4
52 ->54->F4
133->113->E7
248->235->E8
484->481->E9
(?)->980->E10
Example 21 state system 2^10:
a = Table[Flatten[{n/20, N[Flatten[g /. Solve[v[n] + f[n] - 2 + 2*g - 1023 == 0, g]]/480.5]}], {n, 0, 20}];
ListPlot[a, PlotJoined -> True]
The normalized to one Plot has the form of dimension for a multifractal system.
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MATHEMATICA
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Ceiling[Flatten[Table[N[Flatten[g /. Solve[2*Floor[2^(n/2)] - 2 + 2*g - (2^n - 1) == 0, g]]], {n, 1, 32}]]]
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CROSSREFS
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Sequence in context: A027246 A090345 A030270 this_sequence A135019 A017921 A005913
Adjacent sequences: A129754 A129755 A129756 this_sequence A129758 A129759 A129760
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 15 2007
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