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Perrin Sequence
Theinteger sequence defined by the recurrence
 | (1) |
with the initial conditions
,
,
. Thisrecurrence relation is the same as that for thePadovan sequence but with different initial conditions. The first few terms for
, 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEISA001608).
The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply theFibonacci numbers).
is the solution of a third-order linear homogeneousrecurrence equation havingcharacteristic equation
 | (2) |
Denoting theroots of this equation by
,
, and
, with
the unique real root, the solution is then
 | (3) |
Here,
 | (4) |
is theplastic constant
, which is also given by the limit
 | (5) |
The asymptotic behavior of
is
 | (6) |
The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEISA074788), which occur for terms
, 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042, 1214, 1461, 1622, 4430, 5802, 9092, 16260, 18926, 23698, 40059, 45003, 73807, 91405, 263226, 316872, 321874, 324098, ... (OEISA112881), the largest of which areprobable primes and a number of which are summarized in the following table.
 | decimal digits | discoverer | date |
 |  | E. W. Weisstein | Oct. 6, 2005 |
 |  | E. W. Weisstein | May 4, 2006 |
 |  | E. W. Weisstein | Feb. 4, 2007 |
 |  | E. W. Weisstein | Feb. 19, 2007 |
 |  | E. W. Weisstein | Feb. 25, 2007 |
 |  | E. W. Weisstein | Feb. 15, 2011 |
Perrin (1899) investigated the sequence and noticed that if
isprime, then
(i.e.,
divides
). The first statement of this fact is attributed to É. Lucas in 1876 by Stewart (1996). Perrin also searched for but did not find anycomposite number
in the sequence such that
. Such numbers are now known asPerrin pseudoprimes. Malo (1900), Escot (1901), and Jarden (1966) subsequently investigated the series and also found noPerrin pseudoprimes. Adams and Shanks (1982) subsequently found that
is such a number.
See also
Integer Sequence Primes,Padovan Sequence,Perrin Pseudoprime,Plastic Constant,Recurrence Relation SignatureExplore with Wolfram|Alpha
More things to try:
References
Adams, W. and Shanks, D. "Strong Primality Tests that Are Not Sufficient."Math. Comput.39, 255-300, 1982.Amend, B. "FoxTrot.com." Cartoon from Oct. 11, 2005.http://www.foxtrot.com/.Escot, E.-B. "Solution to Item 1484."L'Intermédiare des Math.8, 63-64, 1901.Jarden, D.Recurring Sequences: A Collection of Papers, Including New Factorizations of Fibonacci and Lucas Numbers. Jerusalem: Riveon Lematematika, 1966.Malo, E.L'Intermédiare des Math.7, 281 and 312, 1900.Perrin, R. "Item 1484."L'Intermédiare des Math.6, 76-77, 1899.Sloane, N. J. A. SequencesA001608/M0429,A074788, andA112881 in "The On-Line Encyclopedia of Integer Sequences."Stewart, I. "Tales of a Neglected Number."Sci. Amer.274, 102-103, June 1996.Referenced on Wolfram|Alpha
Perrin SequenceCite this as:
Weisstein, Eric W. "Perrin Sequence."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/PerrinSequence.html