<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=Content-Type content="text/html; charset=iso-8859-1">
<META content="MSHTML 6.00.2600.0" name=GENERATOR></HEAD>
<BODY bottomMargin=0 leftMargin=3 topMargin=0 rightMargin=3>
<DIV></DIV>
<DIV>Jeffrey (and Seqfans),</DIV>
<DIV>> > I wonder if the terms of the above CF has any recurrence
pattern?<BR>> <BR>> Yes: the even-indexed terms (such as 12 = 2*6)
appear to be <BR>> the product of the previous two terms.<BR>> <BR>>
The odd-indexed terms (such as 78=6*12 + 6) appear to be the <BR>> product of
the previous two terms, plus the term two behind.<BR> </DIV>
<DIV>Excellent observation. </DIV>
<DIV>Indeed, note the terms of A112373:</DIV>
<DIV>1,1,2,12,936,68408496,342022190843338960032,<BR>584861200495456320274313200204390612579749188443599552,
...</DIV>
<DIV>where a(n+2) =(a(n+1)^3+a(n+1)^2)/a(n) with a(0)=1,
a(1)=1.<BR> </DIV>
<DIV>It is remarkable to me that the reciprocal sum of these terms </DIV>
<DIV>x = Sum_{n>=0} 1/A112373(n)</DIV>
<DIV>has the continued fraction expansion:</DIV>
<DIV>x =
[2;1,1,2,2,6,12,78,936,73086,68408496,4999703411742,<BR>342022190843338960032,1710009514450915230711940280907486,<BR>584861200495456320274313200204390612579749188443599552,...]<BR> </DIV>
<DIV>such that the even partial quotients of the CF (when offset=0 at constant
term)</DIV>
<DIV>equal A112373 (for n>0):
<BR>2,1,2,12,936,68408496,342022190843338960032,<BR>584861200495456320274313200204390612579749188443599552,...</DIV>
<DIV> </DIV>
<DIV>and the odd partial quotients equal
A112373(n+1)/A112373(n):<BR>1,2,6,78,73086,4999703411742,1710009514450915230711940280907486,...</DIV>
<DIV> </DIV>
<DIV>> It would be nice to prove this. Probably an easy induction,
but</DIV>
<DIV> </DIV>
<DIV>Yes, a proof would be nice; it may indeed have a simple
reason behind it all.</DIV>
<DIV> </DIV>
<DIV>I will submit this CF and the constant to Neil when the time is right.
</DIV>
<DIV>I will include your nice recurrence and credit you with it. </DIV>
<DIV> </DIV>
<DIV>Thanks! </DIV>
<DIV> Paul</DIV></BODY></HTML>