Padovan polynomials

Padovan polynomials

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

P_n(x)=left{begin{matrix}
x,qquadqquadqquadqquad&mbox{if }n=1 1,qquadqquadqquadqquad&mbox{if }n=2 x^2,qquadqquadqquadqquad&mbox{if }n=3 xP_{n-2}(x)+P_{n-3}(x),&mbox{if }nge4 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.

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