[seqfan] sequences based on continued fractions
wouter meeussen
wouter.meeussen at pandora.be
Sun Aug 8 19:57:18 CEST 2010
hi all,
I just submitted 3 new seq and a comment to an existing one, based on the
'special' continued fractions
K[w,n] = w/(1+w/(2+w/(3+...+w/( (n-1) +w/n) ... ))) and
K[n,w] = 1/(w+2/(w+3/(w+4/(w+ ...+ (n-1) /(w+n/w)))))
Each can be written as a quotient of polynomials in n : num(w)/denom(w).
It gives a good feeling to notice that at least one of these four is already
present in the OEIS.
Even better : it turns out to be realy easy to come up with something 'close
to a closed form' for them, based on known sequences. Again, it shows how
important it is to document and 'decorate' each sequence so that it can be
nicely reconstructed.
At this time however, my 'close to closed forms', twice a binomial sum and
twice recurrence formula, hang in there as mere conjectures. They seem to be
linked to 'Lah triangles' and modified Hermite polynomials, but those links
go a bit higher than I can reach. Maybe for someone of taller stature?
Also, the only link that I could find to Ramanujan's conjecture
1/(1+2/(1+3/(1+4/.. = 0.5251352761... = -1+1/(Sqrt(E
Pi/2)-Sum[1/(2k-1)!!,{k,1,Infinity}]
is to a Hungarian discussion forum
http://www.komal.hu/forum/forum.cgi?a=to&tid=94&st=50&dr=1&sp=1233 that
might be temporary. Surely proper literature refs. must exist.
see A180047, A180048, A180049 and A084950 in a few days.
sequentially yours,
Wouter.
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