Journal of Integer Sequences, Vol. 18 (2015), Article 15.8.4

Curious Continued Fractions, Nonlinear Recurrences, and Transcendental Numbers

Andrew Hone
School of Mathematics, Statistics and Actuarial Science
University of Kent
Canterbury CT2 7NF
United Kingdom


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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