Padovan polynomials
Encyclopedia
In mathematics
, Padovan polynomials are a generalization of Padovan sequence
numbers. These polynomial
s are defined by:
The first few Padovan polynomials are:
The Padovan numbers are recovered by evaluating the polynomials Pn-3(x) at x = 1.
Evaluating Pn-3(x) at x = 2 gives the nth Fibonacci number plus (-1)n.
The ordinary generating function for the sequence is
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Padovan polynomials are a generalization of Padovan sequence
Padovan sequence
The Padovan sequence is the sequence of integers P defined by the initial valuesP=P=P=1,and the recurrence relationP=P+P.The first few values of P are...
numbers. These polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s are defined by:
The first few Padovan polynomials are:
The Padovan numbers are recovered by evaluating the polynomials Pn-3(x) at x = 1.
Evaluating Pn-3(x) at x = 2 gives the nth Fibonacci number plus (-1)n.
The ordinary generating function for the sequence is