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

$P_n\left(x\right)=left\left\{begin\left\{matrix\right\}$

The first few Padovan polynomials are:

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

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.