I)for all real k nonzero. Pi=2^(n)*sqrt(2)*sqrt(1/k-2^(-k)*sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+cdots+sqrt(2+sqrt(2))))))))^k/k);.Pi_(9)=2^(9)*sqrt(2)*sqrt(1/k-2^(-k)*sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))))^k/k);.II).For all K >0.Pi=2^(n)*sqrt(2)*sqrt(-(sqrt(3)*sqrt(sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+cdots+sqrt(2+sqrt(2))))))))^(k)*2^(-k)*(6*k^2-4*k)-3*k^2+4*k))/(3*k^2-2*k)+(3*k)/(3*k^2-2*k));.Approx (n=9) .Pi_(9)=2^(9)*sqrt(2)*sqrt(-(sqrt(3)*sqrt(sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))))^(k)*2^(-k)*(6*k^2-4*k)-3*k^2+4*k))/(3*k^2-2*k)+(3*k)/(3*k^2-2*k));.III).For all K <0 . Pi=2^(n)*sqrt(2)*sqrt((sqrt(3)*sqrt(sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+cdots+sqrt(2+sqrt(2))))))))^(k)*2^(-k)*(6*k^2-4*k)-3*k^2+4*k))/(3*k^2-2*k)+(3*k)/(3*k^2-2*k));.Approx (n=9).Pi_(9):2^(9)*sqrt(2)*sqrt((sqrt(3)*sqrt(sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2+sqrt(2))))))))^(k)*2^(-k)*(6*k^2-4*k)-3*k^2+4*k))/(3*k^2-2*k)+(3*k)/(3*k^2-2*k));.IV).Algorithmk in ]0,2] ; .a(n+1)=2^(n-k/4+3/2)*sqrt((3^(5/4)*k*2^(k/2)-sqrt(3)*sqrt((6*k^2-4*k)*(2^((-4*n-3)/k+1)*((3*k^2-2*k)*a(n)^4-3*k*2^(2*n+2)*a(n)^2+3*2^(4*n+3))^(1/k)+2*3^(1/k))^(k/2)+(4*sqrt(3)*k-3^(3/2)*k^2)*2^k))/(3^(5/4)*k^2-2*3^(1/4)*k));.a(2)=2^(5/2)*sqrt((3*k)/(3*k^2-2*k)-(sqrt(3)*sqrt((6*k^2-4*k)/2^(k/2)-3*k^2+4*k))/(3*k^2-2*k));.n --- infinity, a(n) ---- Pi .Gamma and Pi. Pi=sqrt((3*%gamma)/(3*%gamma^2-2*%gamma)-(sqrt(3)*sqrt(((sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(sqrt(2)+2)+2)+2)+2)+2)+2)+2)^(%gamma/2)*(6*%gamma^2-4*%gamma))/2^%gamma-3*%gamma^2+4*%gamma))/(3*%gamma^2-2*%gamma))*2^(n+1/2) ;.