[seqfan] Re: [math-fun] strange approximations for F(x) = partition function. regarding A000041.

[Simon Plouffe]

Hello mr Miller,

  hum, interesting, we surely do not use the same
methods at all, me : I look at the numbers directly,

  I do not see where it connects, the arguments I use
are rationals, he uses algebraic arguments also
the approx. where does it comes from ??, I do not
see it.

  I have found these approximations when I did a series
of mistakes, I first used the non-standard starting point
(n+1) instead of n, starting at 1 and not 0.

Then I used the wrong settings for digits,
and PSLQ bugged, PSLQ has lots of problems with
values that are approximations by the way,
then I used pari-gp, more reliable,  again with the
wrong settings for the precision, and even there
that false identity was still popping out !, I could
not understand the error, this is where I started to
investigate what was the problem, then I found the
one for 2/5 and then later the one for 4/5 and no
other, and still wondering why these values are
so good and unique ??.

  In all this, if we use F(x) the partition function
which is the 'base' function, usualy a property
of that function will propagate to the others
of the same type, eta-dedekind, ramanujan Tau,
theta series, etc. it is all related to the same
base function which is F(x) the partitions. (ref.
G.H. Hardy, collected papers, Vol. V). This is
why I am investigating that one.

  Yesterday, I have found a very simple trick to
compute the partitions and those related functions
with only a good knowledge of some math. constants
like Pi, log(Pi), log(2) and log(GAMMA(1/4)),
with these at hand, I can compute p(n) directly
, I do not use Rademacher formula, the recurrence
formula at all, it is very simple. I am not
saying that my algorithm is faster, it is not.
It is very simple, direct and not
much theory behind actualy. I read Hardy's papers
for years, inspired by the circle method and
all that, followed by the Rademacher formula
and or the recurrence which is efficient but
not simple enough for me.

  best regards, and thank you for that
reference of Van der Poorten et al.

  Simon Plouffe