See A178160 for definition of "type2 Trottlike constant."
Each time a type2 Trottlike constant is extended by one digit, the size of the interval encompassing all the values that can be reached using the terms of an infinitelyextended sequence of positive onedigit numbers directly as digits (e.g., [0.375511111...,0.375599999...]) decreases by a factor of 10.
However, the size of the interval encompassing all the values that can be reached using the terms of such a sequence as terms of a continued fraction of the type used in the second Trott constant (A091694) decreases more slowly.
As a result, as the number of digits increases, the continuedfraction interval increasingly dwarfs the other interval, as is seen in comparing the intervals for a(3)=15751525912:
0+1/(5+7/(5+1/(5+2/(5+9/(1+2/(1+1/(9+9/(1+1/(9+9/...))))))))) = 0.157515259128210971946776833132...
0+1/(5+7/(5+1/(5+2/(5+9/(1+2/(9+9/(1+1/(9+9/(1+1/...))))))))) = 0.157537568512450889006911743529...
which defines an interval whose width is about 0.0000223, while the width of the interval [0.15751525912111...,0.15751525912999...] is only 0.00000000000888....
Inextensible type2 Trottlike constants can occur only when one of the endpoints of the continuedfraction interval happens to fall inside the much smaller interval; this becomes less and less likely as the number of digits increases.
Also, because inextensible cases arise only when increasing the value of a proposed next digit would cause the continued fraction value to increase, inextensible cases must satisfy (k+1) mod 4 < 2, where k is the number of digits to the right of the decimal point. An exhaustive search found no 12digit inextensible cases, so a(4) > 10^14.
