Quite confused about two sequences

[Ralf Stephan]

> "Superseeker" did not have an answer for a(n):
> *******************************
> Report on [ 1,1,6,15,27,36,27,27,162]: 
> 
> [snip]
> Guesss suggests that the generating function F(x) 
> may satisfy the following algebraic or differential equation: 
> 
> x^5-3/7*x^4+2/7*x^3+1/3*x^2-1/9*x+1/18+(5/7*x^4-2/7*x^3-1/6*x^2+1/6*x-1/18)*F(x) = 
> 0 
> 
> If this is correct the next 6 numbers in the sequence are: 
> 
> [729, 13365/7, 21384/7, 12393/7, -210681/49, -65610/7] 

If you get from a program (guesss, guessgf etc) a rational generating
function (or diff. eq.) that produces non-integer further terms then
the program simply has no answer for you. This can be due to several
factors, e.g. not enough terms, default parameters within the program,
or your sequence just does not belong in the C-finite universe. It is
not easy to decide which, in such a case, but it always helps to have
more terms, as any C-finite sequence will show itself sooner or later.

See Zeilberger's article which includes a holonomic systems introduction
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPS/holonomic.ps
(A HOLONOMIC SYSTEMS APPROACH TO SPECIAL FUNCTIONS IDENTITIES)

ralf