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Curious Continued Fractions, Nonlinear Recurrences, and Transcendental Numbers
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Andrew Hone

School of Mathematics, Statistics and Actuarial Science

University of Kent

Canterbury CT2 7NF

United Kingdom

**Abstract:**

We consider a family of integer sequences generated by nonlinear
recurrences of the second order, which have the curious property that
the terms of the sequence, and integer multiples of the ratios of
successive terms (which are also integers), appear interlaced in the
continued fraction expansion of the sum of the reciprocals of the
terms. Using the rapid (double exponential) growth of the terms, for
each sequence it is shown that the sum of the reciprocals is a
transcendental number.

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(Concerned with sequences
A112373
A114550
A114551
A114552.)

Received June 30 2015;
revised version received July 20 2015.
Published in *Journal of Integer Sequences*, July 20 2015.

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