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A178162 Digits (after the initial "0.") of Inextensible Type-2 Trott-like Constants. 1
3755, 48269854, 15751525912 (list; graph; refs; listen; history; text; internal format)



See A178160 for definition of "type-2 Trott-like constant."

Each time a type-2 Trott-like constant is extended by one digit, the size of the interval encompassing all the values that can be reached using the terms of an infinitely-extended sequence of positive one-digit 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 continued-fraction 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 type-2 Trott-like constants can occur only when one of the endpoints of the continued-fraction 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 12-digit inextensible cases, so a(4) > 10^14.


Table of n, a(n) for n=1..3.


3755 is in the sequence because 0.3755 satisfies the definition of a type-2 Trott-like constant, but none of the 5-digit numbers 37551 through 37559 do; i.e., the constant 0.3755 cannot be extended.

To verify, observe that the minimum and maximum values for an infinitely-long continued fraction beginning with 0+3/(7+5/(5+...)) (with no zeros after the initial term) are

0+3/(7+5/(5+1/(9+9/(1+1/(9+9/(1+1/...)))))) = 0.375529994...


0+3/(7+5/(5+9/(1+1/(9+9/(1+1/(9+9/...)))))) = 0.407056182...

respectively, and that the interval [0.375529994...,0.407056182...] intersects the interval [0.375511111...,0.375599999...], but the same is not true of the intervals corresponding to any of the 5-digit numbers 37551 through 37559.


Cf. A091694, A178160.

Sequence in context: A180974 A285158 A238031 * A236051 A133967 A133969

Adjacent sequences:  A178159 A178160 A178161 * A178163 A178164 A178165




Jon E. Schoenfield, May 21 2010



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Last modified July 27 03:49 EDT 2021. Contains 346302 sequences. (Running on oeis4.)