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

0,3

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.

Links

Table of n, a(n) for n=0..104.

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)

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).

Example

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

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 *)

Crossrefs

Sequence in context: A340675 A356497 A097884 * A114233 A279047 A063086

Adjacent sequences: A094815 A094816 A094817 * A094819 A094820 A094821

Keyword

nonn,easy

Author

Guy Valette, Jun 12 2004

Extensions

More terms from David Wasserman, Jun 29 2007

Status

approved