A famous functional equation

That is following problem: Find all functions f:\mathbb{R}\to\mathbb{R} such that
f(x^2 + y + f(y)) = (f(x))^2 + 2\cdot y\; \forall x,y\in\mathbb{R}.(*)
It is famous! Why? Because, it is from AMM(problem 10908, posted by Wu Wei Chao) and it is one in problems from Bulgarian TST 2003, Vietnam TST 2004 and Iran TST 2007. However, in Vietnam TST 2004 it has form:

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Two lemmas on linear sequences of order two

Note: My English is so bad but don’t worry about that! 🙂

A sequence (a_n) is called linear of order two if there are real numbers p,q such that a_{n+2}=pa_{n+1}+qa_n\;\forall n=1,2,3,...

In this topic we’ll use ideas in proofs of two following lemmas to solve some Olympiad problems.

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