[seqfan] Update on A112373
Andrew N W Hone
A.N.W.Hone at kent.ac.uk
Thu Jul 2 14:22:55 CEST 2015
Dear Seqfans,
After I submitted sequence A112373 back in 2005, Paul Hanna made some beautiful empirical observations about
the sum of the reciprocals of the terms,
1+1+1/2+1/12+1/936+... = 2.5844017240...,
namely that the continued fraction expansion of this number has coefficients which are the terms
A112373(n) interlaced with the successive ratios A112373(n+1)/A112373(n). This led Paul to submit the sequences
A114550, A114551, A114552. However, despite some very interesting correspondence with other seqfans including Gerald McGarvey, Mitch Harris and Jeffrey Shallit, it seems that the proof of the original observations was never discussed.
Some time later (I think it was 2009) I thought about this again, and in thinking about the proof I found a family of
sequences which generalizes A112373 and displays an analogous behaviour in terms of the continued fraction of
the sum of reciprocals. I've finally got around to looking at an old notebook and have written this up in a short note:
http://arxiv.org/abs/1507.00063
I've also submitted this to J. Integer Sequences, so if it gets accepted then I will link it to the OEIS.
A nice additional observation is that the number 2.5844017240... (see A114550) turns out to be transcendental, which is a consequence of the fact that A112373 grows so fast.
One could submit many more examples of sequences to the OEIS, based on the construction in my paper, but somehow A112373 seems like the neatest example of its kind.
Best wishes,
Andy
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