close form solution for A136010 (per PURRS http://www.cs.unipr.it/purrs/)
Joerg Arndt
arndt at jjj.de
Sun Mar 30 00:32:24 CET 2008
For those who want to refresh the basics about computations of
recurrences using matrix powers or (better!) powers modulo the
characteristic polynomial, Binet forms, recurrences for subsequences,
Binet forms etc., I dare to point to pp.655-665 of the text
http://www.jjj.de/fxt/#fxtbook
The stuff is written for programmers and hopefully accessible
for people without strong background in math.
Ralf Stephan's ggf (Guess Generating Function) script is
always handy, a copy is here:
http://www.jjj.de/pari/ggf.inc.gp
cheers, jj (feeling a bit like a spammer now)
* Max Alekseyev <maxale at gmail.com> [Mar 30. 2008 10:00]:
> On Sat, Mar 29, 2008 at 11:35 AM, Maximilian Hasler
> <maximilian.hasler at gmail.com> wrote:
>
> > Maple:
> > A136010:=n->simplify((10+61/sqrt(85))*(7/2-1/2*sqrt(85))^n+(10-61/sqrt(85))*(7/2+1/2*sqrt(85))^n);
> >
> > PARI:
> > A136010(n) = round((10+61/sqrt(85))*(7/2-1/2*sqrt(85))^n+(10-61/sqrt(85))*(7/2+1/2*sqrt(85))^n)
>
> The latter is not that good since sqrt() in PARI is not precise but
> floating-point function (in contrast to Maple, where it is symbolic by
> default). Therefore, this function may give wrong results for large n
> or an error like:
>
> *** round: precision too low in truncr (precision loss in truncation).
>
> In PARI it is better to use the following code that does not involve
> any floating point operations:
>
> A136010(n) = local(y=Mod(x,x^2-85));
> lift((10+61/y)*(7/2-1/2*y)^n+(10-61/y)*(7/2+1/2*y)^n)
>
> Regards,
> Max
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