A Recursively-Generated Integer Sequence

[Leroy Quet]

[posted also to sci.math]


I wrote:

> 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


I mention this sequence in the sci.math thread:

http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&safe=off&threadm=glfacbn
6rlg7%40forum.mathforum.com&rnum=2&prev=

I forgot to mention here that
each n(m) is coprime with each n(k), for every m not = k.

thanks,
Leroy Quet