In mathematics, Padovan polynomials are a generalization of

Padovan sequence In number theory, the Padovan sequence is the sequence of integers ''P''(''n'') defined. by the initial values :P(0)=P(1)=P(2)=1, and the recurrence relation :P(n)=P(n-2)+P(n-3). The first few values of ''P''(''n'') are :1, 1, 1, 2, 2, 3, 4, 5 ...

numbers. These

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s are defined by: :

P_n(x) = \begin
1,                     &\mboxn=1\\
0,                     &\mboxn=2\\
x,                     &\mboxn=3\\
xP_(x)+P_(x),&\mbox n\ge4.
\end

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_(x)=4x^3+1\,

P_(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 ''n''th

Fibonacci number In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start ...

plus (−1)^''n''. The

ordinary generating function In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary ser ...

for the sequence is :

\sum_^\infty P_n(x) t^n = \frac{1 - x t^2 - t^3} .

See also