[seqfan] Re: Sum over inverse palindromes

Alexander P-sky apovolot at gmail.com
Thu Oct 21 05:51:52 CEST 2010


> so it seems the value of is sum_{n>=2} 1/A002113(n) = 3.370283259..

8/9-56/(27*Pi)+Pi   ~~=3.370283260023
99/2+163/(2*e)-28*e~~=3.3702832586192

On 10/20/10, Richard Mathar <mathar at strw.leidenuniv.nl> wrote:
>
> http://list.seqfan.eu/pipermail/seqfan/2010-October/006257.html says
>
> ri> Yes, the sum converges.
> ri> The number of n-digit palindromes is O(10^(n/2)) while the reciprocal
> ri> of each is less than 10^(1-n), so the sum of the reciprocals of the
> ri> n-digit palindromes is O(10^(-n/2)).
>
> This is not good. Now I feel obliged to come up with some crude estimates
> of the limit -:). The partial sums over 1/p for palindromes p<= 10^d are
>
> d   convergent
> 2  2.8289682539682539683
> 3  3.0861471861471861472
> 4  3.3190020006509114708
> 5  3.3421302129563524738
> 6  3.3651662022380450003
> 7  3.3674689702500643779
> 8  3.3697715736297805439
> 9  3.3700018324036169535
> 10  3.3702320909393699278
> 11  3.3702551167906460788
> 12  3.3702781426416106403
> 13  3.3702804452267040622
> 14  3.3702827478117970990
>
>
> Wynn's extrapolation for the first 3, 5, 7,... of these values generates
> estimators
> 5.54812148...
> 3.36995258...
> 3.37028506...
> 3.3702832624...
> 3.37028325945240...
> 3.37028325949737...
>
> so it seems the value of is sum_{n>=2} 1/A002113(n) = 3.370283259..
>
> # Yet another maple program (yamp):
> Digits := 20 ;
> # return the value of sum_p 1/p where p are dgs-digits palindromes
> invPali := proc(dgs)
>         local ret,dgshlf,m,p;
>         ret := 0.0 ;
>         dgshlf := floor(dgs/2) ;
>         if type(dgs,'even') then
>                 # if count of digits is even, run through all dgs/2 size
>                 # numbers of the form 100.. up to 999..., flip them
>                 # to generate p, and add 1/p.
>                 for n from 10^(dgshlf-1) to 10^dgshlf-1 do
>                         p := n*10^dgshlf+digrev(n) ;
>                         ret := ret+evalf(1/p) ;
>                 end do:
>         else
>                 # if count of digits is odd, run through all floor(dgs/2)
> size
>                 # numbers of the form 100.. up to 999..., flip them
>                 # insert m from 0 to 9 in the middle to generate p, and add
> 1/p.
>                 for n from 10^(dgshlf-1) to 10^dgshlf-1 do
>                         p := n*10^(1+dgshlf)+digrev(n) ;
>                         for m from 0 to 9 do
>                                 ret := ret+evalf(1/p) ;
>                                 p := p+10^dgshlf ;
>                         end do:
>                 end do:
>         end if;
>         ret ;
> end proc:
> # start with x=sum over the 1-digit palindromes
> x := add(1/n,n=1..9) :
> x := evalf(x) ;
> # accumulate the inverted 2-digits to 14-digit palindromes in x
> for dgs from 2 to 14 do
>         x := x+invPali(dgs) ;
>         print(x) ;
> end do:
>
>
>
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>
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