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

[Jess Tauber]

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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