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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. 1
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; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..72.

Wikipedia, E_7½

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: A213940 A236027 A220128 * A236455 A033093 A070032

Adjacent sequences:  A131495 A131496 A131497 * A131499 A131500 A131501

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Aug 12 2007

STATUS

approved

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Last modified December 12 11:34 EST 2018. Contains 318060 sequences. (Running on oeis4.)