[seqfan] Checking the numerical integration for A355594

[Hugo Pfoertner]

Dear Sequence Fanatics,

In the sequence https://oeis.org/A355954, the correctness of the calculated
constant depends entirely on the function intnum for numerical integration
in PARI being able to calculate an integral with a very high level of
accuracy if the number of digits for the floating-point calculation is set
to 1000 or even higher, for example.

It's about the integral with which the electrical resistance between two
nodes of an infinite triangular grid of one-ohm resistors can be
calculated. It is given, for example, in the comment of
https://oeis.org/A355589 . In the appendix of the article by Atkinson and
Steenwijk linked there, it is also specified as a Mathematica function
Rtri[n_, p_], and it is claimed that Mma can solve this integral for small
values of n and p (or j and k in my case) in closed form, such that the
rational representations given in https://oeis.org/A355585 can be
calculated directly (potentially also using Simplify). That was supposedly
already possible with Mathematica before the year 1999.

My request to those of you who are good at using Mathematica and can also
carry out somewhat more expensive calculations (time, memory requirements)
is as follows:

Calling Mma with the function Rtri(n,p) as given in the attached article on
page 491. Is a current Mma implementation able to directly deliver the
results for small n and p that are in the overview table in A355585?

Does Mma reproduce the specified digits (29 in the current draft) of the
constant A in A355954 when calculating e.g. A = Rtri(20*10^6,0) -
log(20*10^6)/(Pi*sqrt(3))? The calculation in PARI was executed  with
\p1500, i.e. 1500 decimal places (Run time some hours), after checking that
the number of stable digits of A increases monotonically when increasing
the call parameter in a geometric progression with a growth factor of 1.2.

Is it possible to calculate even more digits of the constant with Mma, e.g.
by calculating with 5*10^7, 10^8, ... as arguments instead of 2*10^7? PARI
then fails because the stack size overflows, although I can run it on a
computer with 256 GB of main memory and use almost the entire memory as a
stack. A crazy task would be the calculation of Rtri(118805048562,
33636581266), for which I expect a result of about 5.00..+ 5.6*10^-23 (from
the asymptotic approximation). With PARI I have zero chance of verifying
this result.

Of course, it would be much better than these attempts with brute force if
someone could transform the integration similar to the method shown in
(Cserti 2000) for the square lattice https://oeis.org/A355953 in such a way
that a closed representation of the constant would be found.

I thank you in advance. Have fun with our wonderful database!

Hugo Pfoertner