Padovan polynomials

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

P_n(x)=\left\{\begin{matrix} x,\qquad\qquad\qquad\qquad&\mbox{if }n=1\\ 1,\qquad\qquad\qquad\qquad&\mbox{if }n=2\\ x^2,\qquad\qquad\qquad\qquad&\mbox{if }n=3\\ xP_{n-2}(x)+P_{n-3}(x),&\mbox{if }n\ge4 \end{matrix}\right.

The first few Padovan polynomials are:

P_1(x)=x \,
P_2(x)=1 \,
P_3(x)=x^2 \,
P_4(x)=2x \,
P_5(x)=x^3+1 \,
P_6(x)=3x^2 \,
P_7(x)=x^4+3x \,
P_8(x)=4x^3+1\,
P_9(x)=x^5+6x^2\,

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

Evaluating Pn-1(x) at x = 2 gives the nth Fibonacci number plus (-1)n. (sequence A008346 in OEIS)

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