Fibonacci polynomials

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In mathematics, Fibonacci polynomials are a generalization of Fibonacci numbers. These polynomials are defined by:

F_n(x)=\left\{\begin{matrix} 1,\qquad\qquad\qquad\qquad&\mbox{if }n=1\\ x,\qquad\qquad\qquad\qquad&\mbox{if }n=2\\ xF_{n-1}(x)+F_{n-2}(x),&\mbox{if }n\ge3 \end{matrix}\right.

The first few Fibonacci polynomials are:

F_1(x)=1 \,
F_2(x)=x \,
F_3(x)=x^2+1 \,
F_4(x)=x^3+2x \,
F_5(x)=x^4+3x^2+1 \,
F_6(x)=x^5+4x^3+3x \,

The Fibonacci numbers are recovered by evaluating the polynomials at x = 1.

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