

A131498


For D_2 type groups as polyhedra: {F,V,E,dimension}>{n+2,2*n,3*n,2*n*(2*n1)/2} such that Euler's equation is true: V=EF+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
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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, pages 171173) as the unification level of symmetry. This level appears to be very near the E11 of 482 that Landsberg's equation gives.


LINKS

Table of n, a(n) for n=1..72.
J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), no. 1, pp. 143179.
Wikipedia, E_7½


FORMULA

{a(n),a(n+1),a(n+2),a(n+3)} = {m+2,2*m,3*m,2*m*(2*m1)/2}: m=floor(n/4).
From Luce ETIENNE, Dec 31 2019: (Start)
a(n) = 3*a(n4)  3*a(n8) + a(n12).
a(n) = (n^2 + 10*n + 36 + (n^2  6*n  20)*(1)^n + 2*(n^2 6*n  8)*cos(n*Pi/2)  8*(n4)*sin(n*Pi/2))/32. (End)
G.f.: x*(3 + 2*x + 3*x^2 + x^3  5*x^4  2*x^5  3*x^6 + 3*x^7 + 2*x^8) / ((1  x)^3*(1 + x)^3*(1 + x^2)^3) (conjectured).  Colin Barker, Jan 03 2020


EXAMPLE

D10>{12, 20, 30, 190};
SO(20) has dimension 190 and D10 has the dodecahedron (E8like) polyhedral configuration 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

Cf. A000384, A000027, A005843, A008585.
Sequence in context: A236027 A220128 A324468 * A236455 A033093 A070032
Adjacent sequences: A131495 A131496 A131497 * A131499 A131500 A131501


KEYWORD

nonn,uned


AUTHOR

Roger L. Bagula, Aug 12 2007


STATUS

approved



