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 A131498 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. 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, 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 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

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Last modified September 20 10:51 EDT 2021. Contains 347584 sequences. (Running on oeis4.)