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Volume 10, Issue 2

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T R O T T ' S   C O R N E R
Michael Trott

Existing Sequences

The On-Line Encyclopedia of Integer Sequences currently (January 2006) lists the following Trott constants:

www.research.att.com/projects/OEIS?Anum=A039662
0, 1, 0, 8, 4, 1, 0, 1, 5, 1, 2, 2, 3, 1, 1, 1, 3, 6, 1, 5, 1, 1, 2, 9, 0, ...

Meaning the following holds:

www.research.att.com/projects/OEIS?Anum=A091694
0, 2, 7, 3, 9, 4, 4, 1, 9, 5, 7, 3, 9, 2, 7, 1, 6, 1, 7, 1, 7, 1, 4, 5, 9, ...

So the following holds:

www.research.att.com/projects/OEIS?Anum=A113307
0, 4, 8, 2, 6, 7, 7, 2, 8, 1, 9, 3, 9, 1, 8, 1, 5, 9, 9, 4, 9, ...

Section 1.2.3 of The Mathematica GuideBook for Programming uses a version of the following one-liner to find such a number.

Here is a continued fraction made from this sequence of integers.

Here is the corresponding fraction and its 100-digit decimal approximation.

The first 100 digits of the continued fraction and the decimal expansion are identical.

We will implement an expanded version of the program that calculated heldCF and use it to find various numbers (which MathWorld's creator Eric Weisstein calls "Trott's Constants" [1]) whose simple or nonsimple continued fraction approximation agrees with their base radix representation up to a few hundred digits.



     
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