[seqfan] Re: A family of quadratic recurrences
Jaume Oliver i Lafont
joliverlafont at gmail.com
Sun Oct 4 10:10:08 CEST 2009
Thank you all for your comments, especially the proof and complete
explanation by Andrew Hone.
Andy wrote:
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P.S. There are several ways to generalize (*) to get other recurrences
of a similar type. For example, premultiply the term with \sum_{j<k}
by an arbitrary (integer) parameter, add on a term of the form C
\sum_j x_j for some (integer) parameter C, and add on a constant D to
the left hand side of (*). Mutatis mutandis all the above arguments
apply to such an equation, and will generate families of
double-exponentially growing integer sequences.
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Another generalization that describes many sequences is
a(n)a(n-L)=Sum{i=1,L-1} Sum{j=i,L-1} a(n-i)w(i,j)a(n-j)
Sequences with a(n)=2a(n-1)-a(n-2) have w(1,1)=w(2,2)=-2 and w(1,2)=5 with L=3.
Somos sequences have w(i,j)=1 when i+j=L, and 0 otherwise.
The triangular number N=L(L+1)/2 reflects the form of the triangular
matrix W (here is upper, but can be lower as well) if both sums runned
from 1 to L-1.
Jaume
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