[seqfan] Re: Recursions in decimal expansions

[Charles Greathouse]

So which numbers can be expressed as a power series with integer
numerator and denominator a power of ten? That would address at least
the first part of the sequence.

And what can be said for other linear recurrences?

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Mon, Sep 24, 2012 at 1:19 PM, Simon Plouffe <simon.plouffe at gmail.com> wrote:
>
>
>
> hello every body,
>
>  I made up a formula a while ago,
> for every integer in fact, there is a way to
> expand it into a power series,
>
> for example, 97, 49 , 48, as follows,
>
>                     1/97, "can be expanded with this serie"
>
>                                    infinity
>                                     -----    (n - 1)
>                                      \      3
>                             1/97 =    )     --------
>                                      /        (2 n)
>                                     -----   10
>                                     n = 1
>
>> dec(49);
>                     1/49, "can be expanded with this serie"
>
>                                  infinity
>                                   -----   /   (n - 1)\
>                                    \      |2 2       |
>                           1/49 =    )     |----------|
>                                    /      |   (2 n)  |
>                                   -----   \ 10       /
>                                   n = 1
>
>> dec(98);
>                     1/98, "can be expanded with this serie"
>
>                                    infinity
>                                     -----    (n - 1)
>                                      \      2
>                             1/98 =    )     --------
>                                      /        (2 n)
>                                     -----   10
>                                     n = 1
>
>
> I made a program based on an old formula I had,
>
> ######################################################
> # program for the expansion of 1/p in powers of an integer
> ##############################################################
> ############# In Maple
>
>
>
> dec:=proc(p)
> local n, p1, p2, p3, p4, ex;
>     ex := trunc(log10(p) + 1);
>     p2 := trunc(10^ex/p);
>     p3 := trunc(10^ex) - p*trunc(10^ex/p);
>     p4 := ex*n;
>     print(1/p, "can be expanded with this serie");
>     return 1/p = Sum(p2*p3^(n - 1)/10^p4, n = 1 .. infinity)
> end;
>
>
> and for 9899 I get this formula :
>
> 1/9899, "can be expanded with this serie"
>
>                 infinity
>                  -----      (n - 1)
>                   \      101
>        1/9899 =    )     ----------
>                   /         (4 n)
>                  -----    10
>                  n = 1
>
> Best regards,
>
>  Simon plouffe
>
>
>
>
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