Correction for A052944
Pieter Moree
moree at science.uva.nl
Fri Mar 5 10:42:24 CET 2004
Rainer Rosenthal wrote:
> Please let me forward a correction I received in de.sci.mathematik from
> Hermann Kremer, when I tolf them about A052944.
>
> I think he will be right (as nearly always), but I just wanted to
> ask here.
>
> Best regards
> Rainer Rosenthal
> r.rosenthal at web.de
>
> : ID Number: A052944
> : URL: http://www.research.att.com/projects/OEIS?Anum=A052944 :
> Sequence: 0,2,5,10,19,36,69,134,263,520,1033,2058,4107,8204,16397, :
> 32782,65551,131088,262161,524306,1048595,2097172,4194325, :
> 8388630,16777239,33554456,67108889,134217754,268435483, :
> 536870940,1073741853,2147483678
> : Name: a(n) = 2^n + n - 1
> : G.f.: (-2+3*x)/(-1+2*x)/(-1+x)^2
>
> The G.f. is OK ...
>
> : Recurrence: {a(1)=5,a(0)=2,2*a(n)-a(n+1)+1-n}
>
> But not the recursion, which should read
>
> { a(0)=0, a(1)=2, a(2)=5, a(n+3) = 4*a(n+2) - 5*a(n+1) + 2*a(n) }
>
This is correct of course. It follows from the theory of recurrences
with constant coefficients that the characteristic polynomial for a
recurrence for numbers of the form
c*2^n+(a*n+b), a non-zero, is (x-2)*(x-1)^2=x^3-4x^2+5x-2.
Another recurrence for the
given sequence is clearly: a(n+1)=2*a(n)-n+2 with
a(0)=0.
Pieter Moree
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