mixed linear recurrences
Michael Somos
somos at grail.cba.csuohio.edu
Mon Nov 4 21:02:17 CET 2002
seqfans,
Clark Kimberling asked about sequences with linear
recurrences whose coefficients depend on the parity of
n. I encountered similar sequences in my Somos sequence
research. There are at least two responses to this kind
of situation. One is to look at bisection or decimation
of the original sequence. This will restore the constancy
of the coefficients. The other is to just regard the
coefficients as periodic sequences themselves. This is
a more interesting response. In fact, in general, when a
sequence satisfies a(n+1)=a(n), then you could regard this
as a function with period one instead of a constant sequence.
The issue is a general one. Whenever you have a recursion,
you can consider the case where the recursion varies with
a parameter which is periodic of small period. There are
even more general situations, but they get very complicated.
Shalom, Michael
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