// Workers AI · traducción al español
Polinomios de Padua
En matemáticas, los polinomios de Padua son una generalización de los números de secuencia de Padua.
Traducción generada por IA. El artículo original en inglés continúa abajo.
// Workers AI · dad joke mode What did Padovan polynomials say? "We're poly in love".
From Wikipedia, the free encyclopedia
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}}}.}
