# 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 $$G_+ = \{1,\,\varphi,\,\varphi^2,\,\varphi^3,\,\dots\}$$ and $$G_- = \{1,\,-\varphi^{-1},\,\varphi^{-2},\,-\varphi^{-3},\,\ldots\}$$, where $$\varphi^2 = \varphi + 1$$, $$\varphi > 0$$, are also elements of this sequence.

What makes a basis for this vector space?

×