[seqfan] Re: Duplicated "A*x+B" sequences
Jaume Oliver i Lafont
joliverlafont at gmail.com
Mon Nov 30 20:21:56 CET 2009
With regards to
http://list.seqfan.eu/pipermail/seqfan/2009-November/003082.html,
sequences with
a(n)=p*n+q-a(n-1)
also have
a(n-1)=p*(n-1)+q-a(n-2).
Subtracting these equations, the (simpler?) recurrence
a(n)=p+a(n-2)
is obtained.
Moreover, since
a(n-2)=p+a(n-4),
all of them satisfy the fourth-order linear recurrence
a(n)=2a(n-2)-a(n-4).
This is written as (0,2,0,-1) in the OEIS index for linear recurrences
http://research.att.com/~njas/sequences/Sindx_Rea.html#recLCC
, like sequences
http://research.att.com/~njas/sequences/A006370 and
http://research.att.com/~njas/sequences/A026741
The denominator in the g.f. for these family of sequences divides (1-x^2)^2.
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