[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


  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

  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

More information about the SeqFan mailing list