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The Perrin Sequence is the integer sequence defined by * /P(1)=0/ * /P(2)=2/ * /P(3)=3/ * /P(n)=P(n-2)+P(n-3)/ There is a closed form formula: EQN:P(n)=\alpha^n+\beta^n+\gamma^n where EQN:\alpha EQN:\beta and EQN:\gamma are the solutions to the cubic equation EQN:x^3-x-1=0. This has one real solution and two complex solutions which are complex conjugates, and whose modulus is less than one. Taking EQN:\beta and EQN:\gamma to be the solutions with modulus less than 1, this gives [[[> EQN:\alpha\approx1.32471795976162... ]]] EQN:P(n)\approx\alpha^n, and so the sequence exhibits exponential growth. Since the EQN:n^{th} term is roughly EQN:\alpha^n, we can take log base ten and see that EQN:log_{10}(\alpha)=0.1221234... and so EQN:n^{th} term will have roughly EQN:0.12n decimal digits. The Perrin Sequence has the amazing property that it seems that /n/ divides /P(n)/ if and only if /n/ is a prime number. This conjecture seems solid, certainly holding for /n/ up to 10^5, but it fails for n=271441. This is a great example how patterns fail for large enough cases. Here are the first few values ... COLUMN_START | /n/ | /P(n)/ | Divides | | 1 | 0 | #Yes# | | 2 | 2 | #Yes# | | 3 | 3 | #Yes# | | 4 | 2 | No | | 5 | 5 | #Yes# | | 6 | 5 | No | | 7 | 7 | #Yes# | | 8 | 10 | No | COLUMN_SPLIT | /n/ | /P(n)/ | Divides | | 9 | 12 | No | | 10 | 17 | No | | 11 | 22 | #Yes# | | 12 | 29 | No | | 13 | 39 | #Yes# | | 14 | 51 | No | | 15 | 68 | No | | 16 | 90 | No | COLUMN_SPLIT | /n/ | /P(n)/ | Divides | | 17 | 119 | #Yes# | | 18 | 158 | No | | 19 | 209 | #Yes# | | 20 | 277 | No | | 21 | 367 | No | | 22 | 486 | No | | 23 | 644 | #Yes# | | 24 | 853 | No | COLUMN_END ---- * http://en.wikipedia.org/wiki/Perrin_number * http://mathworld.wolfram.com/PerrinSequence.html * http://www.google.com/search?q=perrin+sequence