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 171-173) as the unification level of symmetry. This level appears to be very near the E11 of 482 that Landsberg's equation gives.

LINKS

J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), no. 1, pp. 143-179.

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).

From Luce ETIENNE, Dec 31 2019: (Start)

a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).

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*(n-4)*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 (E8-like) 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

KEYWORD

nonn,uned

AUTHOR

Roger L. Bagula, Aug 12 2007

STATUS

approved