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In mathematics , Padovan polynomials are a generalization of Padovan sequence numbers. These polynomials are defined by:[ 1]
P
n
(
x
)
=
{
1
,
if
n
=
1
0
,
if
n
=
2
x
,
if
n
=
3
x
P
n
−
2
(
x
)
+
P
n
−
3
(
x
)
,
if
n
≥
4.
{\displaystyle 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
{\displaystyle P_{1}(x)=1\,}
P
2
(
x
)
=
0
{\displaystyle P_{2}(x)=0\,}
P
3
(
x
)
=
x
{\displaystyle P_{3}(x)=x\,}
P
4
(
x
)
=
1
{\displaystyle P_{4}(x)=1\,}
P
5
(
x
)
=
x
2
{\displaystyle P_{5}(x)=x^{2}\,}
P
6
(
x
)
=
2
x
{\displaystyle P_{6}(x)=2x\,}
P
7
(
x
)
=
x
3
+
1
{\displaystyle P_{7}(x)=x^{3}+1\,}
P
8
(
x
)
=
3
x
2
{\displaystyle P_{8}(x)=3x^{2}\,}
P
9
(
x
)
=
x
4
+
3
x
{\displaystyle P_{9}(x)=x^{4}+3x\,}
P
10
(
x
)
=
4
x
3
+
1
{\displaystyle P_{10}(x)=4x^{3}+1\,}
P
11
(
x
)
=
x
5
+
6
x
2
.
{\displaystyle P_{11}(x)=x^{5}+6x^{2}.\,}
The Padovan numbers are recovered by evaluating the polynomials Pn −3x ) at x = 1.
Evaluating Pn −3x ) at x = 2 gives the n th Fibonacci number plus (−1)n (sequence A008346 in the OEIS )
The ordinary generating function for the sequence is
∑
n
=
1
∞
P
n
(
x
)
t
n
=
t
1
−
x
t
2
−
t
3
.
{\displaystyle \sum _{n=1}^{\infty }P_{n}(x)t^{n}={\frac {t}{1-xt^{2}-t^{3}}}.}
