Define A(x) by A(x) = 1 + x / A(x^2), so A(x) = 1+x/(1+x^2/(1+x^4/(1+x^8/(1+ ...)))). Define F(x) by F(x) = 1 + x / (1 - x^2 / F(x^2) ), so F(x) = 1+x/ (1-x^2/ (1+x^2/ (1-x^4/ (1+x^4 /(1-x^8/ (1+x^8/ (1-x^16/ ... ))))))). The following took me a while: Show that F(x) - x = A(x^3) Spoiler below. I am asking because I'd like to know how you'd rate it in difficulty and time this should/did take. Regards, jj ... on to the spoiler ... on to the spoiler ... on to the spoiler ******************** SPOILER ********************** Set G(x) = F(x) - x observe that G(x) = 1 + x^3/G(x^2) = 1 + x^3/(1 + x^6 / G(x^2) ) = ... = 1 + x^3/(1 + x^6 / (1 + x^12 / (1 + x^24 / (...) ) ) ) = A(x^3). Cf. https://oeis.org/draft/A238429 (expansion of F), https://oeis.org/draft/A218031 (expansion of A)