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A131498
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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.
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1
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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
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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{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).
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
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EXAMPLE
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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.
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MATHEMATICA
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a = Table[{n + 2, 2*n, 3*n, 2*n*(2*n - 1)/2}, {n, 1, 20}]; Flatten[a]
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CROSSREFS
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KEYWORD
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nonn,uned
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AUTHOR
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STATUS
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approved
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