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A109441
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A sequence of Cartan group dimensions, sorted and duplicates deleted: Dim{A_n,B_n,C_n,D_n}={(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}.
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1
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0, 1, 3, 4, 6, 8, 10, 15, 16, 21, 24, 28, 35, 36, 45, 48, 55, 63, 64, 66, 78, 80, 91, 99, 100, 105, 120, 136, 143, 144, 153, 168, 171, 190, 195, 196, 210, 224, 231, 253, 255, 256, 276, 300, 324, 325, 351, 378, 400, 406, 435, 465, 484, 576, 676, 784, 900
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OFFSET
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1,3
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COMMENTS
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Result gives new exceptional groups isomorphisms: E_6->D_6 ( 78) E_7 1/2-> D_9 ( 190) E_9(E_11 ?)-> C_10 ( 484) And none for G_2,F_4,E_7,E_8. (14,52,133,248) That reasoning separates the exception groups into two types.
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REFERENCES
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R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.
Todd Rowland, "Dynkin Diagram", http://mathworld.wolfram.com/DynkinDiagram.html
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LINKS
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Table of n, a(n) for n=1..57.
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FORMULA
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A_n->Dim_SL(n+1)=(n+1)^2-1 B_n->Dim_SO(2*n+1)=(2*n+1)*(2*n+1-1)/2 C_n->Dim_SP(2*n)=((2*n)^2 D_n->Dim_SO(2*n)=2*n*(2*n-1)/2 a(n)=Sort[{(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}]
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MATHEMATICA
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Union[Flatten[Table[{(n + 1)^2 - 1, (2*n + 1)*(2*n + 1 - 1)/2, (2*n)^2, (2*n)*(2*n - 1)/2}, {n, 0, 15}]]]
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CROSSREFS
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Cf. A120722.
Sequence in context: A173339 A064269 A120150 * A176986 A069357 A080728
Adjacent sequences: A109438 A109439 A109440 * A109442 A109443 A109444
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KEYWORD
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nonn,uned
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AUTHOR
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Roger L. Bagula, Jun 22 2007
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STATUS
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approved
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