[seqfan] Re: [math-fun] Mathematical hell, sequence and the sequence A007699

[Simon Plouffe]

Hello,

  there are many exotic examples like that.

  Look at this one,

  the sequence A007699, a(n)= {nearest integer to } a(n-1)^2 /a(n-2).

  the sequence is : 10, 219, 4796, 105030, 2300104, 50371117, 
1103102046, ...

  the ratio of a(n+1)/a(n) -->> 21.89949541893233438052532024370859294946
6190194047646611617481379446934015013...
quite rapidly.

  By testing that number for algebraicity, one quickly finds that
it is one of the real roots of

      4       3      2
11 x  - 18 x  + 3 x  - 22 x + 1

, the root can be calculated explicitely but it is quite ugly.

  BUT this is FALSE, the exact number, the ratio of a(n+1)/a(n)
to high precision is :

21.89949541893233438052532024370859294946619019404764661\
     1617481379446934015013222134632935910397445152107249\
     9761349032053359109441276427149926005650693640825950\
     1335437931357406880506038288883182166121707555714731\
     7300725684058890389637589630893096102492114259366306\
     8931460123024765848662425871358354508021664249361199\
     6944188439509332536170556143293330518711985939602460\
     9252047620048792957280978376910436207360514815505096\
...

  Is NOT a simple algebraic the root of the <apparent> equation
is valid up to 0.11357748460267988639402531902 * 10^(-1878).

  As far as I know, nobody really knows WHY it behaves like
that, those Pisot sequences are quite bizarre in my humble
opinion.

  There are others, like A010900, that one too is exotic
enough. E(4,13).

  A good idea would be to test against any asymptotics
the sequence (21.8994954189323343805253202437085929494661)^n
to see how far it is from the real sequence.

  I discussed that topic with David Boyd (UBC) a while ago
and the problem with these animals is that we could have
  a situation where the exponents (or order)
of the recurrence relation could be bery big. That
means that we can have some approximations but the
real thing is far more complicated.

ps : I added this comment on the edit page of
the sequence.

PS2 : this is one example where whatever sophistication
we now have with programs like PSLQ, LLL, Pari and
all that, we still can find examples where it fails
to give a definite answer.

  best regards,
  Simon Plouffe