Padovan polynomials

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

$P_n(x)=\{\begin{array}{cc}1,& \text{if }n=1\\ 0,& \text{if }n=2\\ x,& \text{if }n=3\\ x{P}_{n-2}(x)+P_{n-3}(x),& \text{if }n\ge 4.\end{array}$

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

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

See also

• Polynomial sequences