[seqfan] Re: Can you identify this decimalized expansion?

[Charles Greathouse]

Same as the larger zero of that polynomial, yes.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Mon, Sep 16, 2013 at 3:34 PM, Jess Tauber <yahganlang at gmail.com> wrote:

> Thanks for this- by the way, is this the same as the solution for x + 1/x =
> 2.1 or just close to it?
>
>
> On Sun, Sep 15, 2013 at 11:29 AM, rkg <rkg at cpsc.ucalgary.ca> wrote:
>
> > (21+sqrt(41))/20 = 1.**370156211871642434324410883731**0906632... =
> > [1,2,1,2,2,1,5,1,2,2,1,5,1,2,**2,1,5,...]    R.
> >
> >
> > On Sat, 14 Sep 2013, Jess Tauber wrote:
> >
> >  1.37015621(0).....
> >>
> >> It isn't in OEIS so far as I can tell.
> >>
> >> This comes from the ratio of the decimalized expansions of EVERY OTHER
> >> shallow diagonal from the Pascal Triangle, but with the following
> caveat.
> >>
> >> Normally readings of terms to be decimalized work upwards along the
> >> shallow
> >> diagonal. If we do this the ratio between the decimalized expansions of
> >> each (not every other) such expansion has limit 1/(5+sqrt35).
> >>
> >> But here I've reversed the usual order and direction of the
> >> decimalization,
> >> working downwards through the diagonals.
> >>
> >> Now we end up with TWO different series, one with larger and one with
> >> smaller values (though both continue to grow).
> >>
> >> Taking the sets separately, and ratios between the expansions of every
> >> other keeping us in the individual sets, both converge on the value at
> >> top,
> >> from opposite directions (the ratio of the larger values rising, and the
> >> ratio of the smaller falling).
> >>
> >> Interestingly the two sets have the relationship such that the sum of a
> >> larger plus the previous smaller gives the next larger, but the sum of
> the
> >> smaller plus 1/10th the value of the larger gives the next smaller. I
> used
> >> this procedure to get to the ratio above, rather than knocking myself
> out
> >> with the Pascal Triangle itself (I'm sure since all of you must know how
> >> to
> >> label and calculate any term in the system you could save yourselves
> most
> >> of the work in any case).
> >>
> >> I had *hoped* from the first couple of digits that this might be related
> >> to
> >> the reciprocal of the Fine Structure Constant from physics, no such luck
> >> of
> >> course.
> >>
> >> Does this sucker have some simple expressible fractional value, like the
> >> reversed decimalization does?
> >>
> >> Jess Tauber
> >>
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> >>
> >>
> >>
> >
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