# [seqfan] strange approximations for F(x) = partition function. regarding A000041.

Simon Plouffe simon.plouffe at gmail.com
Wed Feb 23 18:11:30 CET 2011

```Hello,

I stumbled on these 2 approximations regarding F(x),
the partition function, where p(n) = the number
of partitions of n, as usual. aka A000041.

F(x) = sum(p(n)*x^n, n=0..infinity):

Instead I use F*(x) = sum(p(n)*x^(n+1), n=0..infinity):

Then here is the strange thing, for x = exp(-2*Pi/5) then
the value is 1/sqrt(5), well almost ; the precision is
13 digits.

For x = exp(-4*Pi/5) the value is
1/2+3/2/sqrt(5)-sqrt(1/2*(1+3/sqrt(5))) the
precision is 28 decimal digits. I find this quite surprising.
I was sure it was exact, it is NOT. I verified with large values.

Also, apparently these are the only 2 examples I have found
within F60 : the Farey fractions up to denominators = 60.
Also when x = exp(-Pi/5) = apparently nothing algebraic of
a low degree.

caution : do not mistake these values for the standard
F(x) which goes 0 for the exponent too, it is not the
same.

I added these 2 values in the formulas of A000041 of course.

Does anybody have an idea why these values just pop out
like that and apparently no other ???!

Bonne journée.
Simon Plouffe

```