Padovan polynomials

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

$P_{n}(x)={\begin{cases}1,&{\mbox{if }}n=1\\0,&{\mbox{if }}n=2\\x,&{\mbox{if }}n=3\\xP_{n-2}(x)+P_{n-3}(x),&{\mbox{if }}n\geq 4.\end{cases}}$

The first few Padovan polynomials are:

$P_{1}(x)=1\,$

$P_{2}(x)=0\,$

$P_{3}(x)=x\,$

$P_{4}(x)=1\,$

$P_{5}(x)=x^{2}\,$

$P_{6}(x)=2x\,$

$P_{7}(x)=x^{3}+1\,$

$P_{8}(x)=3x^{2}\,$

$P_{9}(x)=x^{4}+3x\,$

$P_{10}(x)=4x^{3}+1\,$

$P_{11}(x)=x^{5}+6x^{2}.\,$

The Padovan numbers are recovered by evaluating the polynomials P_n−3(x) at x = 1.

Evaluating P_n−3(x) at x = 2 gives the nth Fibonacci number plus (−1)ⁿ. (sequence A008346 in the OEIS)

The ordinary generating function for the sequence is

$\sum _{n=1}^{\infty }P_{n}(x)t^{n}={\frac {t}{1-xt^{2}-t^{3}}}.$

See also

• Polynomial sequences