Padovan polynomials

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

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

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)

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

See also