The Perrin Sequence is the integer sequence defined by

There is a closed form formula: $P(n)=\alpha^n+\beta^n+\gamma^n$ where $\alpha$ $\beta$ and $\gamma$ are the solutions to the cubic equation $x^3-x-1=0.$ This has one real solution and two complex solutions which are complex conjugates, and whose modulus is less than one.

Taking $\beta$ and $\gamma$ to be the solutions with modulus less than 1, this gives
$\alpha\approx1.32471795976162...$
$P(n)\approx\alpha^n,$ and so the sequence exhibits exponential growth.

Since the $n^{th}$ term is roughly $\alpha^n,$ we can take log base ten and see that $log_{10}(\alpha)=0.1221234...$ and so $n^{th}$ term will have roughly $0.12n$ 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 ...
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
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
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

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