The Fibonacci sequence 0,1,1,2,3,5,... is an object of different modifications and extensions. In [1,2] four different ways of constructing two sequences {α i } ∞ i=0 and {β i } ∞ i=0 are described and called 2-Fibonacci sequences (or 2-F-sequences). The four schemes are the following α 0 = a, β 0 = b, α 1 = c, β 1 = d α n+2 = β n+1 + β n , n ≥ 0 β n+2 = α n+1 + α n , n ≥ 0 α 0 = a, β 0 = b, α 1 = c, β 1 = d α n+2 = α n+1 + β n , n ≥ 0 β n+2 = β n+1 + α n , n ≥ 0 α 0 = a, β 0 = b, α 1 = c, β 1

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... d α n+2 = β n+1 + α n , n ≥ 0 β n+2 = α n+1 + β n , n ≥ 0 α 0 = a, β 0 = b, α 1 = c, β 1 = d α n+2 = α n+1 + α n , n ≥ 0 β n+2 = β n+1 + β n , n ≥ 0 Obviously, the Third and the Fourth schemes contain two standard Fibonacci sequences and therefore they are trivial modification, while the first two schemes are essential extensions of Fibonacci sequence. Graphically, the (n+2)-nd members of the four schemes are obtained from the n-th and the (n+1)-st members as shown in Fig. 1-4. β n m β n+1 m β n+2 m α n m α n+1 m α n+2 m d d d d d ' Fig. 1. 25