Binet-like Formula
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:
where a, b and c are constants.
Since the magnitudes of the complex roots q and r are both less than 1 (and hence p is a Pisot–Vijayaraghavan number), the powers of these roots approach 0 for large n, and tends to zero.
For all, P(n) is the integer closest to, where s = p/a = 1.0453567932525329623... is the only real root of s3 − 2s2 + 23s − 23 = 0. 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.
Read more about this topic: Padovan Sequence
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