page_title: Perrin number
section_title: Binet-like formula
hierarchical_section_title: Perrin number / Properties / Binet-like formula
caption_attribution_description: English: Spiral of equilateral triangles with side lengths which follow the Perrin sequence. Akin to File:Padovan triangles (1).png. 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29... The first two terms may be ignored or the spiral may be considered to begin clockwise (3, 0, 2), and then flip to counterclockwise (2, 3, 2...), with both 3s in the same location. The 3 triangle is marked dark blue to point out it being snubbed.
context_page_description: In mathematics, the Perrin numbers are defined by the recurrence relation
  P = P + P for n > 2,
  with initial values
  P = 3, P = 0, P = 2.
  The sequence of Perrin numbers starts with
  3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, ...
  The number of different maximal independent sets in an n-vertex cycle graph is counted by the nth Perrin number for n > 1.
context_section_description: The Perrin 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 Perrin sequence analogue of the Lucas sequence Binet formula is
  Since the magnitudes of the complex roots q and r are both less than 1, the powers of these roots approach 0 for large n. For large n the formula reduces to
  This formula can be used to quickly calculate values of the Perrin sequence for large n. The ratio of successive terms in the Perrin sequence approaches p, a.k.a. the plastic number, which has a value of approximately 1.324718. This constant bears the same relationship to the Perrin sequence as the golden ratio does to the Lucas sequence. Similar connections exist also between p and the Padovan sequence, between the golden ratio and Fibonacci numbers, and between the silver ratio and Pell numbers.
caption: Spiral of equilateral triangles with side lengths which follow the Perrin sequence.
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