[seqfan] Re: Recursions in decimal expansions

[Robert Munafo]

I wrote a lot about these types of fractions at my page "Fractions
with Special Digit Sequences":

  http://www.mrob.com/pub/math/seq-digits.html

Not all are reciprocals or even rational fractions, and I wrote a lot
about generating functions and how to derive one from the other. But
there is a lot that is applicable to A216407, and the three existing
terms (49, 97, 9899) are all mentioned.

On 9/24/12, Charles Greathouse <charles.greathouse at case.edu> wrote:
> A216407, "Positive integers n such that the initial part of the
> decimal expansion of  1/n reveals a recursive sequence", is an
> interesting idea for a sequence.

I think "recurrence relation" is a better phrase, rather than
"recursive sequence". (Povolotsky's original name for A216407 was "...
decimal expansion of [...] 1/m reveals clear pattern of integer
sequence", and then it was changed to the current "... decimal
expansion of  1/n reveals a recursive sequence".)

> I'm bringing this up to SeqFan
> because I don't know how to make it precise. In fact I'm concerned
> that, when made precise, it will turn out to be a duplicate of
> A000027.

I too have difficulty seeing how we could make a precise definition
for this. It's a nice idea, but where do you draw the line? 1/8
includes the first 2 terms of the powers of 2 (0.12...) for the same
reason that 1/98 gives the first 6 terms (0.010204081632...) but does
that mean 1/8 should be included?

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  Robert Munafo  --  mrob.com
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