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A129757 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. 0
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; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..32.

FORMULA

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}]]]

EXAMPLE

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.

MATHEMATICA

Ceiling[Flatten[Table[N[Flatten[g /. Solve[2*Floor[2^(n/2)] - 2 + 2*g - (2^n - 1) == 0, g]]], {n, 1, 32}]]]

CROSSREFS

Sequence in context: A303587 A151524 A030270 * A135019 A141685 A017921

Adjacent sequences:  A129754 A129755 A129756 * A129758 A129759 A129760

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, May 15 2007

STATUS

approved

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Last modified November 17 08:39 EST 2019. Contains 329217 sequences. (Running on oeis4.)