The real vector space of "Fibonacci-like" sequences contains all sequences satisfying for all natural numbers . 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 and the Lucas number are both elements of this sequence. The two geometric series and , where and are also elements of this sequence.
What makes a basis for this vector space?