The above definition is the one given by Ian Stewart and by MathWorld. Other sources may start the sequence at a different place, in which case some of the identities in this article must be adjusted with appropriate offsets.
In the spiral, each triangle shares a side with two others giving a visual proof that
the Padovan sequence also satisfies the recurrence relation
Starting from this, the defining recurrence and other recurrences as they are discovered,
one can create an infinite number of further recurrences by repeatedly replacing by
The Perrin sequence satisfies the same recurrence relations as the Padovan sequence, although it has different initial values.
The Perrin sequence can be obtained from the Padovan sequence by the
Extension to negative parameters
As with any sequence defined by a recurrence relation, Padovan numbers P(m) for m<0 can be defined by rewriting the recurrence relation as
Starting with m = −1 and working backwards, we extend P(m) to negative indices:
Sums of terms
The sum of the first n terms in the Padovan sequence is 2 less than P(n + 5), i.e.
Sums of alternate terms, sums of every third term and sums of every fifth term are also related to other terms in the sequence:
For example, for k = 12, the values for the pair (m, n) with 2m + n = 12 which give non-zero binomial coefficients are (6, 0), (5, 2) and (4, 4), and:
The Padovan sequence numbers can be written in terms of powers of the roots of the equation
This equation has 3 roots; one real root p (known as the plastic number) and two complex conjugate roots q and r. Given these three roots, the Padovan sequence can be expressed by a formula involving p, q and r :
For all , P(n) is the integer closest to . Indeed, is the value of constant a above, while b and c are obtained by replacing p with q and r, respectively.
The ratio of successive terms in the Padovan sequence approaches p, which has a value of approximately 1.324718. This constant bears the same relationship to the Padovan sequence and the Perrin sequence as the golden ratio does to the Fibonacci sequence.
P(n) is the number of ways of writing n + 2 as an ordered sum in which each term is either 2 or 3 (i.e. the number of compositions of n + 2 in which each term is either 2 or 3). For example, P(6) = 4, and there are 4 ways to write 8 as an ordered sum of 2s and 3s:
2 + 2 + 2 + 2 ; 2 + 3 + 3 ; 3 + 2 + 3 ; 3 + 3 + 2
The number of ways of writing n as an ordered sum in which no term is 2 is P(2n − 2). For example, P(6) = 4, and there are 4 ways to write 4 as an ordered sum in which no term is 2:
4 ; 1 + 3 ; 3 + 1 ; 1 + 1 + 1 + 1
The number of ways of writing n as a palindromic ordered sum in which no term is 2 is P(n). For example, P(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2:
6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1
The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is equal to P(n − 5). For example, P(6) = 4, and there are 4 ways to write 11 as an ordered sum in which each term is odd and greater than 1:
11 ; 5 + 3 + 3 ; 3 + 5 + 3 ; 3 + 3 + 5
The number of ways of writing n as an ordered sum in which each term is congruent to 2 mod 3 is equal to P(n − 4). For example, P(6) = 4, and there are 4 ways to write 10 as an ordered sum in which each term is congruent to 2 mod 3:
A spiral can be formed based on connecting the corners of a set of 3-dimensional cuboids.
This is the Padovan cuboid spiral. Successive sides of this spiral have lengths that are
the Padovan numbers multiplied by the square root of 2.
Erv Wilson in his paper The Scales of Mt. Meru observed certain diagonals in Pascal's triangle (see diagram) and drew them on paper in 1993. The Padovan numbers were discovered in 1994. Paul Barry (2004) showed that these diagonals generate the Padovan sequence by summing the diagonal numbers.