Non-obvious basis

Algebra Level 4

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?

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