[math-fun] Help with some math

Let f(x)^2 = f(x^2) - 1 and assume f is of the form f(x) = x + a x^-1 + b x^-3 + c x^-5 + ... What are the first few coefficients (say 20) as rational numbers? With the tools I have, it would take me a while to power through this. I assume it would be easier with a symbolic math program.