MATL, 15 bytes
17Lt_Qwhi^d5X^/ Try it online! Or verify all test cases.
Explanation
17L % Push ϕ (predefined literal) t_Q % Duplicate, negate, add 1: gives 1-ϕ wh % Swap, concatenate horizontally: gives [1-ϕ, ϕ] i^ % Input n, element-wise power: gives [(1-ϕ)^n, ϕ^n] d % Consecutive difference(s): gives ϕ^n - (1-ϕ)^n 5X^ % Push 5, square root / % Divide.: gives (ϕ^n - (1-ϕ)^n) / sqrt(5) % Implicit display