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

[rkg]

(21+sqrt(41))/20 = 1.3701562118716424343244108837310906632... = 
[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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