# [seqfan] Re: A family of quadratic recurrences

A.N.W.Hone A.N.W.Hone at kent.ac.uk
Thu Oct 1 11:53:46 CEST 2009

```Hi seqfans -

The fact that these sequences consist entirely of integers follows from Theoreom 1.10 in
Fomin and Zelevinsky's article "The Laurent Phenomenon", Advances in Applied Mathematics,
Volume 28 ,  Issue 2  (February 2002) Pages: 119 - 144  which is available on the arxiv at

http://arxiv.org/PS_cache/math/pdf/0104/0104241v1.pdf

These recurrences satisfy the Laurent property, which means that the iterates are
polynomials in the initial values (say a(0),a(1),...,a(L-1)) with integer coefficients.
So setting all L initial values to 1 gives an integer sequence.

These particular ones may also be related to generalized Hurwitz equations, studied by Baragar, Zagier
and others, but I'd need to check that.

Andy

________________________________________
From: seqfan-bounces at list.seqfan.eu [seqfan-bounces at list.seqfan.eu] On Behalf Of Charles Greathouse [charles.greathouse at case.edu]
Sent: 30 September 2009 20:24
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: A family of quadratic recurrences

These double-exponential sequences are hard to work with!  I
calculated L < 5 and L > 9 to 35 terms and 5 <= L <= 9 to 30 terms to
verify that they are integers.  Some of the terms had hundreds of
millions of digits.  I suppose I could extend these with L additional
terms by working mod the last L terms.

On the slightly-related subject of other double exponentials (looked
up to compare to this sequence): A165421 is a duplicate of A011764
(differing only in offset).  Should this stay or be merged?  If
merged, should be recurrence be translated?
%F A011764 a(n) = 3 * a(1) * ... * a(n-1) for n > 1.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Wed, Sep 30, 2009 at 1:31 PM, Jaume Oliver i Lafont
<joliverlafont at gmail.com> wrote:
> Hello Seqfans,
>
> In the family of quadratic recurrences defined by
> a(n)=sum(i=1,L-1,a(n-i)*sum(j=i,L-1,a(n-j)))/a(n-L), with L initial ones,
> I have not been able to find any noninteger value.
>
> Do these recurrences yield only integers? For any L>=2?
>
> This search is related to sequence
> http://research.att.com/~njas/sequences/A165896,
> which is the case L=4.
>
> Regards,
> Jaume
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

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