The integer sequence defined by the recurrence relation
|
(1)
|
with the initial conditions . This is the same recurrence relation as for
the Perrin sequence, but with different initial
conditions.
The recurrence relation can be solved explicitly, giving
|
(2)
|
where
is the
th
root of
|
(3)
|
Another form of the solution is
|
(4)
|
where
is the
th
root of
|
(5)
|
The first few terms are 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, ... (OEIS A000931).
The first few prime Padovan numbers are 2, 2, 3, 5, 7, 37, 151, 3329, 23833, ... (OEIS A100891), corresponding to indices ,3, 4, 5, 7, 8, 14, 19, 30, 37, 84,
128, 469, 666, 1262, 1573, 2003, 2210, 2289, 4163, 5553, 6567, 8561, 11230, 18737,
35834, 44259, 536485, ... (OEIS A112882). The
search for prime numerators has been completed up to
by E. W. Weisstein (Apr. 10, 2011), and
the following table summarizes the largest known values.
| decimal digits | discoverer | |
| 536485 | 65518 | E. W. Weisstein (May 16, 2009) |
| 727734 | 88874 | E. W. Weisstein (Apr. 7, 2011) |
The ratio
|
(6)
|
where
denotes a polynomial root, is called the plastic
constant.
A matrix analogous to the Fibonacci Q-matrix exists for Padovan numbers. Defining
|
(7)
|
the powers of
give
|
(8)
|
(J. Lien, pers. comm., Mar. 11, 2005).