[seqfan] Odd behavior in a sequence
David Wilson
davidwwilson at comcast.net
Wed Oct 21 01:16:32 CEST 2015
I'm looking at the linear recurrence
0, 1, 3, 11, 39, 139, .
with a(n) = 3a(n-1) + 2a(n-2) for n >= 2.
I was interested in the question, for which primes p do {a(n)} == Z (mod p)?
That is, for which primes p do all residues r appear in {a(n)} (mod p)
(Forgive my notational abuse)?
We'll shorten that to "a covers p".
I expected this to have a simple answer, and it almost does.
It turns out that whether or not a covers p most of the time depends on p
mod 17.
In general, a covers p if p == 0, 3, 5, 6, 7, 10, 11, 12, or 14 (mod 7).
However, there seem to be a smattering of primes p:
683, 1217, 2731, 11299, 48817, .
which, according to their residue mod 17, should be covered by a, but are
not.
Very curious.
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