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A094818 Number of classes of dp-homogeneous spherical curves with n double points. 0


%S 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,

%T 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,

%U 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

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

%C 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.

%H Guy Valette, <a href="http://projecteuclid.org/euclid.bbms/1457560855">Double point-homogeneous spherical curves</a>, Bull. Belg. Math. Soc. Simon Stevin 23 (2016), 73-86.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1,0,0,0,1,0,1)

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

%F 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).

%e 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 lima├žon). - _Guy Valette_, Feb 21 2017

%t 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 *)

%K nonn,easy

%O 0,3

%A _Guy Valette_, Jun 12 2004

%E More terms from _David Wasserman_, Jun 29 2007

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