Perrin-type sequence?

[Dean Hickerson]

Mostly to Russell Walsmith:

> Playing around with ordered triplets based on C6, I found a sequence S
> where, iff n is prime, it divides S[n]. This works for 100 terms, anyway,
> which is as far as my coding chops take me for the nonce.
...
> More info in appendix II at
> http://ixitol.com/Triternions.pdf

Your sequence is the absolute value of the sequence a(n) defined by
a(1)=0, a(2)=3, a(3)=5, and a(n)=3a(n-1)-6a(n-2)+8a(n-3).  More
explicitly,

    a(n) = (2^(n-1) - ((1+sqrt(-15))/2)^n - ((1-sqrt(-15))/2)^n)/3

It does appear to be true that if n>3 is prime then n divides a(n); at
least it's true for all n up to 200000.  But the converse is false; the
counterexamples less than 10000 are n = 385, 665, 899, 989, 1045, 1105,
2209, 2345, 3599, 4081, 4187, 6479, 8569.

Dean Hickerson
dean at math.ucdavis.edu