CF of Reciprocal Sum of A112373

[Jeffrey Shallit]

> From: "Paul D Hanna" <pauldhanna at juno.com>
> To: seqfan at ext.jussieu.fr
> Cc: anwh at kent.ac.uk
> Subject: CF of Reciprocal Sum of A112373
> 
> Seqfans,
>        This may or may not be surprising, but I find it interesting neve=
> rtheless.
> The sum of the reciprocal terms of Andrew Hone's sequence A112373:
>  x Sum_{n>0} 1/A112373(n) =
> 
>      2.584401724019776724812076147...
> where A112373 is defined by: =
> 
> a(n+2) = (a(n+1)^3+a(n+1)^2)/a(n) with a(0)=1, a(1)=1 =
> 
>  =
> 
> is a constant with an interesting Continued Fraction:
>  =
> 
> x = [2; 1, 1, 2, 2, 6, 12, 78, 936, 73086, 68408496, 4999703411742,
>  342022190843338960032, 1710009514450915230711940280907486, 584861200495=
> 456320274313200204390612579749188443599552,...]
>  =
> 
> I wonder if the terms of the above CF has any recurrence pattern?

Yes:  the even-indexed terms (such as 12 = 2*6) appear to be 
the product of the previous two terms.

The odd-indexed terms (such as 78=6*12 + 6) appear to be the product
of the previous two terms, plus the term two behind.

It would be nice to prove this.  Probably an easy induction, but
I don't have time right now.  Very beautiful.

Jeffrey Shallit