[seqfan] Re: Duplicated "A*x+B" sequences
Richard Mathar
mathar at strw.leidenuniv.nl
Mon Nov 30 21:05:55 CET 2009
http://list.seqfan.eu/pipermail/seqfan/2009-November/003087.html
jol> Return-Path: <seqfan-bounces at list.seqfan.eu>
jol> Date: Mon, 30 Nov 2009 20:21:56 +0100
jol> From: Jaume Oliver i Lafont <joliverlafont at gmail.com>
jol> To: seqfan at list.seqfan.eu
jol> Subject: [seqfan] Re: Duplicated "A*x+B" sequences
jol> ..
jol> sequences with
jol> a(n)=p*n+q-a(n-1)
jol> also have
jol> a(n-1)=p*(n-1)+q-a(n-2).
jol> ..
jol> Subtracting these equations, the (simpler?) recurrence
jol> a(n)=p+a(n-2)
jol> is obtained.
jol>
jol> Moreover, since
jol> a(n-2)=p+a(n-4),
jol> all of them satisfy the fourth-order linear recurrence
jol> a(n)=2a(n-2)-a(n-4).
jol>
jol> This is written as (0,2,0,-1) in the OEIS index for linear recurrences
jol> http://research.att.com/~njas/sequences/Sindx_Rea.html#recLCC
The recurrences in these cases are of third order after
elimination of any constant term, as demonstrated in
A168195, A168202 or A168212.
So the entries in Sindx_Rea.html#recLCC would refer to the
form equivalent to the denominator (1+x)(1-x)^2 with a third order
degree of the polynomial, namely (1,1,-1), not to the 4-th order (0,2,0,-1) , because
the index associates sequences with the recurrences of smallest polynomial degree.
RJM
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