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

[Jess Tauber]

OK- so for normal decimal expansions for the shallow Pascal diagonal the
limit ratio between diagonals is 1/(5+sqrt35), while for Pascal columns the
limit ratio is 1/(5+sqrt15).

Systemically the former appears to relate to the limit ratio of the Pascal
rows, 1/11 (or 1/(5+sqrt36) and the latter to the limit ratio of the Pascal
edge-parallel diagonals, 1/9 (or 1/(5+sqrt16).

If we square 1/10 the value of the shallow ratio we add 0.1,to that value,
while squaring 1/10 the value of the column ratio we subtract 0.1 to that
value. Some kind of symmetry?

So here's the question- will the limit ratio value for every other
reversed-order decimalized expansion for columns be related to that for the
shallow diagonals? Say, x - 1/x = 2.1? That is 2.5 vs. 0.4? Or something
completely different?


On Tue, Sep 17, 2013 at 9:45 AM, rkg <rkg at cpsc.ucalgary.ca> wrote:

> 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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