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 P_n-1(x) at x = 2 gives the nth Fibonacci number plus (-1)ⁿ.

See also

• Polynomial sequences
• Padovan sequences