In mathematics, Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:[1]
Pn(x)=\begin{cases} 1,&ifn=1\\ 0,&ifn=2\\ x,&ifn=3\\ xPn-2(x)+Pn-3(x),&ifn\ge4. \end{cases}
The first few Padovan polynomials are:
P1(x)=1
P2(x)=0
P3(x)=x
P4(x)=1
| 2 | |
| P | |
| 5(x)=x |
P6(x)=2x
| 3+1 | |
| P | |
| 7(x)=x |
| 2 | |
| P | |
| 8(x)=3x |
| 4+3x | |
| P | |
| 9(x)=x |
P10(x)=4x3+1
P11(x)=x5+6x2.
The Padovan numbers are recovered by evaluating the polynomials Pn−3(x) at x = 1.
Evaluating Pn−3(x) at x = 2 gives the nth Fibonacci number plus (−1)n.
The ordinary generating function for the sequence is
| infty | |
| \sum | |
| n=1 |
Pn(x)tn=
| t | |
| 1-xt2-t3 |
.