Re: [math-fun] Re^2: FW: A conjectural series for 1/pi of a new type

27 Jan 2011

      Le 2011-01-27 13:31, Bill Gosper a Ã©crit :
...
1, 1, 33, 97, 1729, 8001, 105441, 627873,
      6989697, 48363649, 488206753,
      3701949153, 35289342529,
      283146701761, 2610495177057,
      21695983405857, 196218339243777,
      1667338615773441, 14917038493453089,
      128562758660255073,
      1143482133220664769,
      9946278255903268929,
      88205310329762729697,
      771946983805271894433,
      6837125121111415598721
I used GFUN, with guessgf
[1, 1, 33, 97, 1729, 8001, 105441, 627873, 6989697, 48363649,

     488206753, 3701949153, 35289342529, 283146701761,

     2610495177057, 21695983405857, 196218339243777,

     1667338615773441, 14917038493453089, 128562758660255073,

     1143482133220664769, 9946278255903268929,

     88205310329762729697, 771946983805271894433,

     6837125121111415598721]
...
guessgf(%,x);
                             1
                   [--------------------, ogf]
                                   2 1/2
                    (1 - 2 x - 63 x )
It is very efficient,

usually a dozen terms is enough to detect
any kind of formulas of that type.

There is another way when we suspect that
the sequence has an algebraic generating function
and it uses LLL or PSLQ, I made an article about
that method a while ago : the paper is here :
http://arxiv.org/pdf/0912.0072

Simon Plouffe