|
Search: id:A131498
|
|
|
| A131498 |
|
For D_2 type groups as polyhedrons: {F,V,E,dimension}->{n+2,2*n,3*n,2*n*(2*n-1)/2} such that Euler's equation is true: V=E-F+2. |
|
+0
2
|
|
| 3, 2, 3, 1, 4, 4, 6, 6, 5, 6, 9, 15, 6, 8, 12, 28, 7, 10, 15, 45, 8, 12, 18, 66, 9, 14, 21, 91, 10, 16, 24, 120, 11, 18, 27, 153, 12, 20, 30, 190, 13, 22, 33, 231, 14, 24, 36, 276, 15, 26, 39, 325, 16, 28, 42, 378, 17, 30, 45, 435, 18, 32, 48, 496, 19, 34, 51, 561, 20, 36, 54, 630
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
This sequence which has n=2 tetrahedron,n=4 cube, n=10 Dodecahedron seems to be very closely related to the exceptional groups in geometric terms. It seems to answer how E8 and E71/2 are related as well. E8*E8 or SO(32) has dimension 496->{18, 32, 48, 496} which is given in Gribbin's book ( The Search for Superstrings, Symmetry and the Theory of Everything, page 171-173)as the unification level of symmetry. This level appears to be very near the E11 of 482 that Landsberg's equation gives.
|
|
REFERENCES
|
Landsberg, J. M. Manivel, L. The sextonions and E7 1/2, Adv. Math. 201 (2006), no. 1, 143--179. http://en.wikipedia.org/wiki/E7%C2%BD_%28Lie_algebra%29
|
|
FORMULA
|
{a(n),a(n+1),a(n+2),a(n+3) = {m+2,2*m,3*m,2*m*(2*m-1)/2}: m=Floor[n/4]
|
|
EXAMPLE
|
D10->{12, 20, 30, 190}
SO(20) has dimension 190 and D10 has the dodecahedron ( E8 like) polyhedral configuration of: V=12, F=20, E=30
E7 1/2 also has dimension 190.
|
|
MATHEMATICA
|
a = Table[{n + 2, 2*n, 3*n, 2*n*(2*n - 1)/2}, {n, 1, 20}]; Flatten[a]
|
|
CROSSREFS
|
Sequence in context: A082391 A046818 A106584 this_sequence A033093 A070032 A165026
Adjacent sequences: A131495 A131496 A131497 this_sequence A131499 A131500 A131501
|
|
KEYWORD
|
nonn,uned
|
|
AUTHOR
|
Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Aug 12 2007
|
|
|
Search completed in 0.002 seconds
|
