Stephen Lucas in his publications presented individual identities of
integral approximations to Pi with nonnegative integrands for the first
few n=3,4,5,6,7
(see
http://www.austms.org.au/Publ/Gazette/2005/Sep05/Lucas.pdf https://www.researchgate.net/publication/267998655_Integral_approximatio ns_to_p_with_nonnegative_integrands
http://web.maths.unsw.edu.au/~mikeh/webpapers/paper141.pdf
)
I was able to find some parameters for each case presented by Lucas -
such that they satisfy my following conjectured generalization (parameterization)
(-1)^n\cdot(\pi - \text{A002485}(n)/\text{A002486}(n))
=(|i|\cdot2^j)^{-1} \int_0^1
\big(x^l(1-x)^{2(j+2)}(k+(i+k)x^2)\big)/(1+x^2)\; dx
where integer n serves as the index for terms in OEIS A002485(n) and A002486(n),
and {i, j, k, l} are some integers parameters (found experimentally for
each case represented by the specific value of n >= 2).
Based on his calculations results, Thomas Baruchel, in helping me,
found that above parameterization yields infinite number of {i, j, k,
l} solutions for each n >= 2.
Thomas shared with me his calculations results and supplied me with
quite a few of valid combinations of i, j, k, l values - so now I have
a lot of experimentally found five-tuples {n,i, j, k, l}, which satisfy
above parameterization, where n varies in the range from 2 to 26.
Based on this data, of course, it would be nice to find how i, j, k, l
are inter-related between each other and with "n" - but such
inter-relation is not obvious and difficult to derive just by
observation .... (though it is clearly seen that an absolute value of
"i" is strongly increasing as "n" is growing from 2 to 26).
So at this stage I am looking for advise how I should proceed in
processing this data in attempt to find relationship between those
parameters.
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