Let a(n), n=0,1,2,... be a sequence of positive integers all of which
are greater than zero with the possible exception of a(0) which may be
zero.
Let h(-3)=0 , h(-2)=0, h(-1)=1, and for n>-1 let
h(n) = a(n) h(n-1)+h(n-2)+h(n-3)
and similarly let
k(-3)=1, k(-2)=0, k(-1)=0 and for n>-1 let
k(n)= a(n) k(n-1)+ k(n-2)+ k(n-3).
This is similar to one of the ways of defining continued fractions.
I have two questions.
Is it true in general that the ratio r(n)= h(n)/ k(n) converges for
any choice of the sequence a(n) as described above?
If the sequence a(n) is eventually periodic, is it then true that r(n)
converges to a root of a third degree polynomial?
I believe that the answer to each question is yes but have been unable
to prove it. Anyone out there know how to prove these conjectures?
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