Brachylog, 1616 14 bytes
0|0;1⟨t≡+⟩ⁱhℕ↙1;0⟨t≡+⟩ⁱhℕ↙.! Takes input through the output variable and outputs through success or failure, and in the case of success the input variable is unified with 01.
0| The output variable is 0, or 0;11;0 startingStarting with [0[1,1]0], ⁱ iterating ⟨ ≡ ⟩ replacing ⟨t ⟩ the first element of the pair with the last element ⟨ +⟩ and the last element with the sum of the pair h until the first element ℕ↙ is greater than or equal to . the output variable, ! and stopping then, h the first element of the pair is equal to the output variable. ℕ↙.! is necessary for it to terminate on false test cases, and 0| is necessary for it to succeed with an input of 0.