Non-obvious basis

The real vector space of "Fibonacci-like" sequences contains all sequences $\{a_n\}_{n \ge 0}$ satisfying $a_n + a_{n+1} = a_{n+2}$ for all natural numbers $n$. Two sequences can be added by adding their corresponding terms, and a sequence can be multiplied by a scalar by multiplying all elements in the sequence by a real number. (Note that the resulting sequence in each case still satisfies the Fibonacci property and is therefore still in the vector space.)

The Fibonacci numbers $F = \{0,\, 1,\, 1,\, 2,\, 3,\, 5,\, \ldots\}$ and the Lucas number $L = \{2,\,1,\,3,\,4,\,7,\,\ldots\}$ are both elements of this sequence. The two geometric series $G_+ = \left\{1,\,\varphi,\,\varphi^2,\,\varphi^3,\,\dots\right\}$ and $G_- = \left\{1,\,-\varphi^{-1},\,\varphi^{-2},\,-\varphi^{-3},\,\ldots\right\}$, where $\varphi^2 = \varphi + 1$ and $\varphi > 0$ are also elements of this sequence.

What makes a basis for this vector space?