mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 exa ...

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 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 the sequence from ...

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 series ...

for the sequence is :

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

See also