[seqfan] Pi Day 2019: A Discrepancy on A006139?

[Brad Klee]

Hi Seqfans,

Last year in July we noticed a connection between
A006139 and the integral on page 12 of Frits Beukers'
"A Rational Approach to Pi" [1].

Knowledge of the p-recurrence associated to the integral
allows rapid production of the approximant series. A
numerical calculation of the asymptotic quality then
reveals a potential discrepancy. The empirical value
seems to be at least a point lower than the reported
value, as per my comment from July 19, 2018.

Of course, Frits Beukers is a well respected expert,
so it is plausible that a resolution of this discrepancy
depends on a better asymptotic theory. If this is the
case in truth, it would be nice to have another
comment on A006139 indicating so.

Until then, the burden of proof is upon the asserter.

Cheers,

Brad

[1] https://dspace.library.uu.nl/handle/1874/26398

For more detailed calculations, see also:

[2] http://demonstrations.wolfram.com/ApproximatingPiWithTrigonometricPolynomialIntegrals/
[3] https://github.com/bradklee/IntegralDiffEQ