A092053: Value of Continued Fraction [1;1/2,1/3,1/4,...,1/n,...]

[Gerald McGarvey]

It looks like 2n-th convergent of
[1; 1, 1, 1/2, 1/2, 1/3, 1/3, 1/4,..., 1/n, 1/n,...]
= 1/n + Sum_{i=1..n} 1/i^2

The n-th convergent of this CF multiplied by:
(floor(n/2)!)^2 if n is even and by
(floor(n/2)!)^2 * (n+1)/2 if n is odd
appears to be A098738(n) which is
(a(1) = 1, a(2) = 2, a(3) = 2, a(4) = 3, for n >= 3, a(n+2) = a(n+1) + 
a(n)*floor(n/2)*ceiling(n/2).)

Regards,
Gerald

At 05:41 PM 8/12/2006, Gerald McGarvey wrote:
>Seqfans,
>
>(*) Pi/2 = [1; 1, 1/2, 1/3, 1/4,..., 1/n,...]
>gives the same denominators and numerators as Wallis's approximation to 
>Pi/2, see
>http://www.research.att.com/~njas/sequences/A001901
>http://www.research.att.com/~njas/sequences/A001902
>http://mathworld.wolfram.com/WallisFormula.html
>
>Regarding (**), it looks like every other denominator (1,4,36,144,...) of the
>convergents for this CF is a denominator of a partial sum of 
>Sum_{n=1..infinity} 1/n^2
>and that the ratios of the corresponding numerators approaches 1.
>numerators:
>1, 2, 3, 7, 19, 61, 29, 241, 1169, 5989, 5869, 5969, 41183, 291881, 
>288731, 1165949, 128461, 10483741, 10413181, 2095337, 22921699, 253408769, 
>252244529, 253311749
>denominators:
>1, 1, 2, 4, 12, 36, 18, 144, 720, 3600, 3600, 3600, 25200, 176400, 176400, 
>705600, 78400, 6350400, 6350400, 1270080, 13970880, 153679680, 153679680, 
>153679680
>
>-- Gerald
>
>At 04:18 AM 8/10/2006, Paul D. Hanna wrote:
>>Seqfans,
>>       Thanks, Joseph, for your comments.
>>Note that
>>(*) Pi/2 = [1; 1, 1/2, 1/3, 1/4,..., 1/n,...]
>>is equivalent to
>>Pi/2 = 1 + 1/(1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 +... ))))).
>>
>>Now it seems like I've seen this expression before ...
>>Can anyone provide a reference?
>>
>>Notice the similarity between (*) and the following:
>>
>>(**) Pi^2/6 = [1; 1, 1, 1/2, 1/2, 1/3, 1/3, 1/4,..., 1/n, 1/n,...]
>>
>>  = 1 + 1/(1 + 1*1/(1 + 1*2/(1 + 2*2/(1 + 2*3/(1 + 3*3/(1 +... ))))).
>>
>>This (**) is conjecture, but is also most likely true and well-known.
>>I wonder if the convergents of (**) are as interesting ...
>>
>>Would appreciate any references that confirm (*) or (**).
>>Thanks,
>>      Paul