A000003 and singular values

[Joerg Arndt]

It seems that (2 * A000003) gives the degree of the minimal polynomial
of (k_n)^2 where k_n is the n-th singular value, i.e.
K(sqrt(1-k_n^2)/K(k_n)==sqrt(n) (and K is the elliptic function
of the first kind:  K(x) := hypergeom([1/2,1/2],[1], x^2).

Can someone shed some light on this one?

Also, when setting K3(x)=hypergeom([1/3,2/3],[1], x^3)
and solving for x such that K3((1-x^3)^(1/3))/K3(x)==sqrt(n),
then the degree of the minimal polynomial of x^3 is every
third term of A000003, or so it seems.