[seqfan] Re: Conjectures stated in Clark Kimberling's A192750

[Neil Sloane]

Richard, Yes, that's the reasoning I used. And that;s why I changed the
definition to use the pair of recurrences. With that starting point the
rest of the analysis is routine.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Wed, Dec 16, 2015 at 10:27 AM, Richard J. Mathar <mathar at mpia-hd.mpg.de>
wrote:

> Concerning:
> http://list.seqfan.eu/pipermail/seqfan/2015-December/015852.html
>
> njas> A192750 has one clearly labeled conjecture, and two other formulas
> which
> njas> were not labeled as conjectures, but probably were.
>
> If we insert the expression of d_n into the recurrence for c_n and
> vice versa, we get two "inhomogeneous" recurrences for d_n and
> for c_n, where the additional terms are polynomials in the index.
> This falls clearly into the realm of formulas for generating
> functions with rational polynomials, because the generating
> function of the extra term is already a rational polynomial.
> See Eq. (4) in
> http://www.mpia.de/~mathar/public/mathar20071126.pdf
>
> From there the only remaining task is to find the few coefficients
> in numerator and denominator of the generating function.
>
> The Grunewald formulas look like just the Binet-formulas
> associated with these.
>
> Richard Mathar
>
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