[seqfan] Can you identify this decimalized expansion?

[Jess Tauber]

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