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%I #19 Jan 03 2020 10:42:46
%S 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,
%T 10,16,24,120,11,18,27,153,12,20,30,190,13,22,33,231,14,24,36,276,15,
%U 26,39,325,16,28,42,378,17,30,45,435,18,32,48,496,19,34,51,561,20,36,54,630
%N For D_2 type groups as polyhedra: {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.
%C 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.
%H J. M. Landsberg and L. Manivel, <a href="http://dx.doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), no. 1, pp. 143-179.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/E7%C2%BD_%28Lie_algebra%29">E_7½</a>
%F {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).
%F From _Luce ETIENNE_, Dec 31 2019: (Start)
%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12).
%F 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)
%F 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
%e D10->{12, 20, 30, 190};
%e SO(20) has dimension 190 and D10 has the dodecahedron (E8-like) polyhedral configuration V=12, F=20, E=30.
%e E7 1/2 also has dimension 190.
%t a = Table[{n + 2, 2*n, 3*n, 2*n*(2*n - 1)/2}, {n, 1, 20}]; Flatten[a]
%Y Cf. A000384, A000027, A005843, A008585.
%K nonn,uned
%O 1,1
%A _Roger L. Bagula_, Aug 12 2007
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