A130896 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.

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, 28, 105, 17, 17, 32, 136, 19, 19, 36, 171, 21, 21, 40, 210, 23, 23, 44, 253, 25, 25, 48, 300, 27, 27, 52, 351, 29, 29, 56, 406, 31, 31, 60, 465, 33, 33, 64, 528, 35, 35, 68, 595, 37, 37

Offset

1,1

Comments

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.

References

David M. Bishop, Group Theory and Chemistry,Dover Publications, 1993, table 3-7.1, page 46

Links

Table of n, a(n) for n=1..70.

Formula

{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]

Conjectures from Colin Barker, Jan 02 2020: (Start)

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).

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

(End)

Mathematica

a = Table[{2*n + 1, 2*n + 1, 4*n, (2*n + 1)*((2*n + 1) - 1)/2}, {n, 1, 32}]; Flatten[a]

Crossrefs

Cf. A131498.

Sequence in context: A123708 A350501 A102302 * A254279 A029882 A163523

Adjacent sequences: A130893 A130894 A130895 * A130897 A130898 A130899

Keyword

nonn

Author

Roger L. Bagula, Aug 22 2007

Status

approved