[seqfan] Recursions in decimal expansions

[Charles Greathouse]

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'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.

The examples from the sequence explain themselves pretty well:

a(1)=49 is in the sequence, because 1/49 = 0.020408163265306122448...
shows the coefficients 1, 2, 4, 8, 16, 32, ... and then it gets
scrambled.
a(2)=97  is in the sequence, because 1/97 = 0.01030927835051546...
shows the coefficients 1, 3, 9, 27, ... and then it gets scrambled.
a(3)=9899 is in the sequence, because 1/9899 =
0.0001010203050813213455904636832003232649762602283058... shows the
first Fibonacci numbers  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... and
then it gets scrambled.

This is an exhaustive list at present.

The first two examples are simple power relationships, of course
equivalent to Sum_{n >= 1} (2/100)^n = 1/49 and Sum_{n >= 1} (3/100)^n
= 3/97. It seems reasonable to search for these relationships by
looking at the sums of (a/10^b) for b >= 1 and 0 < a < 10^b, then
looking at the fractions with a power of a as numerator. Some examples
not present in the sequence sprint to mind: 1/9 corresponds to the
powers of 1, for example.

But broadening the definition to all linear recurrences to accommodate
9899 makes it harder to think of search strategies. Is every unit
fraction expressible as Sum_{n >= 1} S_n/10^kn for some integer k and
linear recurrence S_1, S_2, ...? If not, how can such unit fractions
be identified?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University