Padovan polynomials

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

:

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 P_"n"-1("x") at "x" = 2 gives the "n"th Fibonacci number plus (-1)^"n". OEIS|id=A008346

ee also

*Polynomial sequences
*Padovan sequences