CF of Reciprocal Sum of A112373

[A.N.W.Hone]

Dear Gerald,

Your remarks concerning the continued fraction related to A112373 are very
interesting to me. Since these require that the ratio of succesive terms
should be integers, and also you have later observed that the ratios of
the ratios should also be integers, I shall outline the proofs of these
facts.

Consider the 2nd order recurrence

x_{n+1} x_{n-1} = x_n^M p(x_n)

where M\geq 2 is an integer and p
is a polynomial with integer coefficients and p(0) \neq 0.
[In my original posting to OEIS I didn't specify M\geq 2; the case M=1
is special - only certain polynomials p will work.]

Let R be the ring of Laurent polynomials in the initial data x_0,x_1, with
integer coefficients, that is R=Z[x_0^{\pm 1},x_1^{\pm 1}] (actually, we
could extend this ring by the coefficients of p).

Then it is easy to prove by induction that x_n \in R, x_{n+1}/x_n \in R
(ratios) and x_{n+1}x_{n-1}/(x_n^2)\in R (ratios of ratios) for all n\in Z.
Therefore, if x_0=1=x_1 then x_n \in Z for all n.

This property of the recurrence is known as the Laurent property
in Fomin and Zelevinsky's work on cluster algebras.

The sequence A112373 is the case M=2, p(y)=y+1.

The CFs of (y^m +y^2)/x mentioned below correspond to the case M=2,
p(y) = y^{m-2}+1.

It would be nice to know that Euler already understood all this!

All the best,
Andy Hone

P.S. For M=1 things are different. For example, the recurrence

x_{n+1} x_{n-1} = x_n^3 + x_n

with x_0=x_1=1 generates a sequence of integers (which I suggest should also
appear in the OEIS), but the ratios of succesive terms are not integers.



On Sat, 10 Dec 2005, Gerald McGarvey wrote:

>
> The ratios of successive terms in A112373 is 1, 2, 6, 78, 73086,
> 4999703411742, 1710009514450915230711940280907486, ...
> If these are all integers (I believe they are), then A112373 is a series of
> the form s = - (x/a - x^2/(ab) + x^3(abc) - x^4/(abcd) ...) where in this
> case x = -1, and so a formula by Euler can be applied, cf. page 31 of
> 'On the Transformation of Infinite Series to Continued Fractions'  by
> Leonhard Euler
> Translated by Daniel W. File
> http://www.math.ohio-state.edu/~sinnott/ReadingClassics/continuedfractions.pdf
>
> Starting with a(0)=.5 instead of a(0)=1 the CF is
> [3, 3, 1, 4, 4, 20, 80, 1620, 129600, 209953620, 27209989152000,
> 5712835722623340193620, ...]
> With a(0)=2 the CF is
> [3, 11, 1, 5, 1, 1, 2, 1, 1, 77, 1, 1, 233, 1, 1, 73085, 1, 1, 17102123, 1,
> 1, 4999703411741, ...]
>
> Notes about CFs for some other related series...
>
> For m > 2, the CFs for (y^m+y^2)/x look related.
> (used x=1.;y=1.;s=1/x+1/y; for(n=1,10, z=(y^m+y^2)/x; x=y; y=z;
> s=s+1/z;);contfrac(s) )
> e.g.
> m=4: [2, 1, 1, 4, 2, 200, 20, 321602000, 80200,
> 53353214040377436196491224020000, 2068556928401602000,...]
> m=5: [2, 1, 1, 8, 2, 23328, 36, 767667136004966560896, 30233736,
> 11428202543809917040939553535324671293831057914382382411094812574379924574115058355844062625584214528,...]
> m=6: [2, 1, 1, 16, 2, 10690688, 68,
> 87818313168587795575060012891154023826432, 49433743624,...]
> m=7: [2, 1, 1, 32, 2, 20037321216, 132,
> 39297387807850612366968992508128169281051499010505152979708234780909568,...]
>