A Recursively-Generated Integer Sequence

[Leroy Quet]

[posted also to sci.math]


This sequence has multiple definitions.

I began wondering about the sequence today using this definition:

n(1) = 1;

n(m+1) is the sum of the numerator and denominator in the (reduced) 
rational:

sum{k=1 to m} 1/n(k).


{n(j)} ->  1, 2, 5, 27, 739, ...


This is sequence A057438, without the initial term, of the Encyclopedia 
of Integer Sequences, which gives the definition: 

>"a(1) = 1; a(n+1) = product_{k = 1 to n} [a(k)] *sum_{j = 1 to n} [1/a(j)]"

({a(j}) -> 1, 1, 2, 5, 27, 739,...)

a(j+1) does in-fact = n(j), since both are defined by the same recursion:

n(m+2) = (product{k=1 to m} n(k)) + n(1+m)^2

= n(m) *(n(1+m) - n(m)^2) + n(1+m)^2.


But what I am wondering is, what is the sum (which does converge)

x = sum {k=1 to oo} 1/n(k)

equal to??



(One more thing: 1 + x 
=
limit{m-> oo}   n(1+m)/(n(2+m)-n(m)^2).)


Anything anyone can add to this discussion??

thanks,
Leroy Quet