[seqfan] Re: Recursions in decimal expansions

[israel at math.ubc.ca]

If positive integer r = sum_{j=0}^m d_j b^j with integers d_j and d_0 > 0, 
then 1/r = sum_{j=0}^infty a_j b^{-j} where sum_{j=0}^m d_j a_{n+j} = 0. 
Thus your example 9899 = -1 - 100 + 100^2 so 1/9899 = sum_{j=0}^infty a_j 
100^j with -a_j - a_{j+1} + a_{j+2} = 0 (the Fibonacci recursion).

If b is a power of 10 and the a_j are nonnegative, we see the pattern in 
the digits of 1/r as long as a_j < b, and they get "scrambled" if a_j > b.

Robert Israel
University of British Columbia


On Sep 24 2012, Charles Greathouse 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'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
>
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