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

rkg rkg at cpsc.ucalgary.ca
Tue Sep 17 15:45:46 CEST 2013


Dear Jess,
           (21+sqrt(41))/20 and its reciprocal, (21-sqrt(41))/20  are the 
roots of the equation you give.  I arrived at them via

x = 1/(1+1/(2+1/y)), where  4y^2 - 5y - 1 = 0.   R.

On Mon, 16 Sep 2013, Jess Tauber 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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