Non-obvious basis

Algebra Level 3

The real vector space of "Fibonacci-like" sequences contains all sequences {an}n0\{a_n\}_{n \ge 0} satisfying an+an+1=an+2a_n + a_{n+1} = a_{n+2} for all natural numbers nn. 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,}F = \{0,\, 1,\, 1,\, 2,\, 3,\, 5,\, \ldots\} and the Lucas number L={2,1,3,4,7,}L = \{2,\,1,\,3,\,4,\,7,\,\ldots\} are both elements of this sequence. The two geometric series G+={1,φ,φ2,φ3,}G_+ = \left\{1,\,\varphi,\,\varphi^2,\,\varphi^3,\,\dots\right\} and G={1,φ1,φ2,φ3,}G_- = \left\{1,\,-\varphi^{-1},\,\varphi^{-2},\,-\varphi^{-3},\,\ldots\right\}, where φ2=φ+1\varphi^2 = \varphi + 1 and φ>0\varphi > 0 are also elements of this sequence.

What makes a basis for this vector space?

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