On Creating A More Generalized Trott's Constant

[Hans Havermann]

Ed Pegg Jr. mentioned a Trott Constant variant in his 21 December  
MathPuzzle web addition. By way of Eric Weisstein's 'Trott's  
Constant' article in MathWorld, the original Trott's Constant dates  
to 1999 and is featured in Sloane's A039662.

http://mathpuzzle.com/
http://mathworld.wolfram.com/TrottsConstant.html
http://www.research.att.com/projects/OEIS?Anum=A039662

Eric states: "It appears to be unique, and all attempts to find other  
such numbers have failed. However, attempts to extend the number of  
digits have proved problematic." The OEIS reference to Simon Plouffe  
gives the number as 0.0108410151... (as opposed to Weisstein's  
0.108410151...) and notes that, in terms of uniqueness, "there may be  
others." A039662 gives 54 terms but has the digits agreeing to only  
"32 places" with Neil stating: "I don't know if this can be modified  
to give more terms of agreement."

I attempted to re-create Trott's Constant in Mathematica in order to  
glean how an algorithm might do the dirty work of extending it and  
had a quick go at manually extending the known terms. Apart from  
recognizing how simple continued fraction expansions that allow zeros  
can be "collapsed" by summing the digits on either side of the zero,  
I didn't make much headway. Reversing the procedure is, of course,  
what allows one to convert a legitimate simple continued fraction  
into a zeros-allowed variant, and I was surprised how easy it was to  
(manually) extend digits simply by using a ContinuedFraction[#] on  
the decimal and feeding back subsequent terms (replacing any greater  
than 9 by their 9090...0x equivalents).

A short foray into a slightly differing 0.108400231... didn't get  
very far but reinforced the already out-there hunch that there exist  
other numbers that mimic Trott's Constant. A quick brute-force search  
for seed-numbers, starting points for Trott-Constant-like numbers,  
resulted in a bunch of candidates. Picking one of these at random, I  
quickly expanded that seed into:

0.3206224113341909061429590611321719090442241121812211423121211112194812 
7129021611212329042902168315212111214..

The continued fraction [0; 3, 2, 0, 6, 2, 2, 4, 1, 1, 3, 3, 4, 1, 9,  
0, 9, 0, 6, 1, 4, 2, 9, 5, 9, 0, 6, 1, 1, 3, 2, 1, 7, 1, 9, 0, 9, 0,  
4, 4, 2, 2, 4, 1, 1, 2, 1, 8, 1, 2, 2, 1, 1, 4, 2, 3, 1, 2, 1, 2, 1,  
1, 1, 1, 2, 1, 9, 4, 8, 1, 2, 7, 1, 2, 9, 0, 2, 1, 6, 1, 1, 2, 1, 2,  
3, 2, 9, 0, 4, 2, 9, 0, 2, 1, 6, 8, 3, 1, 5, 2, 1, 2, 1, 1, 1, 2, 1,  
4] = 2841811577511303983409363256028316730642 /  
8863421510947433793512472278725472500639 which is:

0.3206224113341909061429590611321719090442241121812211423121211112194812 
7129021611293319604592075099759507890..

... giving us 81 decimal digits of agreement but, again, no obvious  
methodology with which to get more. I would guess that the reason  
digit-extension bogs down (eventually) is because of the interplay of  
how these numbers are created with the inherent (in)accuracy afforded  
by each system of number representation.

Perhaps the above is enough to motivate someone more mathematically- 
minded than I to create other examples and figure this thing out.

http://chesswanks.com/sg.html