import math

y0 = math.sqrt(2) - 1
a0 = 6 - (4 * math.sqrt(2))


def f(y):
    fy= (1 - (y**4)) ** (0.25)
    return fy


def calculate_y_kp1(k):
    return (1 - f(y[k])) / (1 + f(y[k]))


def calculate_a_kp1(k):
    return (a[k] * (1 + y[k+1]) ** 4) - (2 ** (2*k+3) * y[k+1] * (1 + y[k+1] + (y[k+1] ** 2)))



y = [y0]
a = [a0]
est = []
for k in range(15):
    y.append(calculate_y_kp1(k))
    a.append(calculate_a_kp1(k))
    est.append(1/a[k+1])

print(est)