

A129757


Maximum genus of fixed edge 2^m1 binary state graph with 2*m+1 states: Vertices(n)=Floor[2^(n/2)]; Faces(n)=Floor[2^[mn/2]; Edges(n)=Vertices(n)+Faces(n)2+2*g=2^m1; 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



