[seqfan] Re: Linear recurrences with large gaps

[Joerg Arndt]

* Ron Hardin <rhhardin at att.net> [Jan 11. 2011 19:21]:
> Running the series for the three recurrences out to n=998 shows that the 
> existing recurrences work, at least out that far.
> 
> Is that far enough to exclude the further term guess?
> 

Hard to say:

? p=subst(p,a,x)+O(x^200)
-1 + 2*x - x^2 + x^81 - 2*x^82 + x^83 + O(x^200)
? printprod(ser2prod(p))  \\ uses 'y' as symbol
  1 * (1-y^1)^{2} * (1-y^81)^{1}  \\ I should have noticed this earlier!

That might be
  (1-y) * prod( n=1, oo, 1-y^(81^n) )
which for oo == 2 is 
? (1-y) * prod( n=0, 2, 1-y^(81^n) ) + O(y^9999)
1 - 2*y + y^2 - y^81 + 2*y^82 - y^83 - y^6561 + 2*y^6562 - y^6563 + y^6642 - 2*y^6643 + y^6644 + O(y^9999)

[but for now this is all just blind guessing]

>  rhhardin at mindspring.com
> rhhardin at att.net (either)
>