oeis at keldesign.com oeis at keldesign.com
Mon Jul 23 04:20:12 CEST 2012

I feel a little selfish talking about my sequence, but I thought of a few
unanswered questions regarding it. I also want to thank T. D. Noe for
simplifying my Mathematica code and attaching the first 1000 terms. The
sequence is defined by:

a(n) = (a(n-1) + a(n-3))/gcd(a(n-1), a(n-3)) with a(0) = a(1) = a(2) =
1.

The starting values 1,1,1 were chosen to be the same as Narayana's Cows
(A000930), but obviously any other initial integers can be chosen (except
0,k,0).

1)       Starting values of 2,2,2 lead to an infinite sequence of 2's, but
this is the only triple that I can find that clearly repeats. Are there
other initial values that repeat?

2)       The sequence starting with 1,1,1 appears to increase indefinitely.
Does it, or does it eventually repeat? Are there any initial values that can
be proven to increase indefinitely? This may not be hard to prove, but I
haven't been able to.

3)       If any of these sequences can be proven to increase indefinitely,
are they exponential, or sub-exponential (lim n->infinity, log(a(n))/n ->
0)?

4)       The sequence starting with 1,1,1 drops to 6 at a(48). Can it ever
get that low again? Is there a lower bound (as a function of n)?

Reed Kelly