%I #7 Jan 02 2020 08:23:40
%S 3,3,4,3,5,5,8,10,7,7,12,21,9,9,16,36,11,11,20,55,13,13,24,78,15,15,
%T 28,105,17,17,32,136,19,19,36,171,21,21,40,210,23,23,44,253,25,25,48,
%U 300,27,27,52,351,29,29,56,406,31,31,60,465,33,33,64,528,35,35,68,595,37,37
%N For D_n type groups as polyhedra that are pyramid-like: {F,V,E,dimension}->{2*n+1,2*n+1,2*n,(2*n+1)*((2*n+1)-1)/2} such that Euler's equation is true: V=E-F+2.
%C This idea comes from the Octahedron being truncated by one vertex to give a pyramid. In this case the n=1 is not a 3d polyhedron, but the rest are very simple F=E figures. Adding one vertex below the plane of the major polygon gives an D_nh type point group ( D_n like figures A131498): these figures are Point groups C_nv.
%D David M. Bishop, Group Theory and Chemistry,Dover Publications, 1993, table 3-7.1, page 46
%F {a(n),a(n+1),a(n+2),a(n+3) = {2*m+1,2*m+1,4*m,(2*m+1)*((2*m+1)-1)/2}: m=Floor[n/4]
%F Conjectures from _Colin Barker_, Jan 02 2020: (Start)
%F G.f.: x*(3 + 3*x + 4*x^2 + 3*x^3 - 4*x^4 - 4*x^5 - 4*x^6 + x^7 + x^8 + x^9) / ((1 - x)^3*(1 + x)^3*(1 + x^2)^3).
%F a(n) = 3*a(n-4) - 3*a(n-8) + a(n-12) for n>12.
%F (End)
%t a = Table[{2*n + 1, 2*n + 1, 4*n, (2*n + 1)*((2*n + 1) - 1)/2}, {n, 1, 32}]; Flatten[a]
%Y Cf. A131498.
%K nonn
%O 1,1
%A _Roger L. Bagula_, Aug 22 2007
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