A094818 Number of classes of dp-homogeneous spherical curves with n double points.

1, 1, 2, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4, 2, 3, 2, 3, 2, 3, 2, 4, 2, 2, 2, 4

Offset: 0

Author

Guy Valette, Jun 12 2004

Keywords

• nonn
• easy

Comments

A spherical curve C is said to be dp-homogeneous if the stability group of C in the group of diffeomorphisms of the sphere acts transitively on the set of double points of C. Two spherical curves belongs to the same class if there is a diffeomorphism of the sphere sending the first curve onto the second one.

Examples

			The second term of the sequence means that all double point-homogeneous spherical curves with just one double point belong to the same orbit relatively to the group of diffeomorphisms of the sphere (it is not true for plane curves: a lemniscate of Bernoulli is not equivalent with a Pascal's limaçon). - _Guy Valette_, Feb 21 2017
		

Links

• Guy Valette, Double point-homogeneous spherical curves, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), 73-86.
• Index entries for linear recurrences with constant coefficients, signature (0,-1,0,0,0,1,0,1)

Programs

Mathematica

CoefficientList[Series[-(x^10 + x^9 + 3 x^8 + 3 x^7 + 5 x^6 + 4 x^5 + 6 x^4 + 3 x^3 + 3 x^2 + x + 1)/(x^8 + x^6 - x^2 - 1), {x, 0, 120}], x] (* Michael De Vlieger, Feb 21 2017 *)

Formula

If n>14, then a(n) = a(n-12).

G.f.: -(x^10+x^9+3*x^8+3*x^7+5*x^6+4*x^5+6*x^4+3*x^3+3*x^2+x+1) / (x^8+x^6-x^2-1).

Extensions

More terms from David Wasserman, Jun 29 2007