[seqfan] Re: A117963

[Harvey P. Dale]

Paul:
  I have tried to implement that formula in Mathematica but it doesn't
generate the terms of the sequence.  Any number, mod 3, cannot exceed 2
but there are many terms in the sequence which exceed that number.  What
don't I understand?
  Best,
  Harvey

-----Original Message-----
From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Paul D
Hanna
Sent: Tuesday, July 24, 2012 11:19 PM
To: seqfan at list.seqfan.eu
Subject: [seqfan] Re: A117963

Harvey, 
     The second and third formulae are equivalent; thus it suffices to
show only that: 
  a(n) == Fibonacci(n+1) (mod 3). 
 
Given the g.f.  
  A(x) = A(x^3)*(1 - 4*x^3 - x^6)/(1 - x - x^2), suppose we define F(x)
such that
  F(x) = F(x^3)*(1 - x^3 - x^6)/(1 - x - x^2), then it is not hard to
see that
  A(x) == F(x) (mod 3).  
 
But now F(x) is simply
  F(x) = 1/(1 - x - x^2)
which is the g.f. for the Fibonacci sequence (with offset). 
 
Therefore the formulae hold. 
 
Best wishes, 
    Paul
---------- Original Message ----------
From: "Harvey P. Dale" <hpd1 at nyu.edu>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Subject: [seqfan] A117963
Date: Tue, 24 Jul 2012 15:20:44 -0400

           I think the 2nd and 3rd formulae provided by Paul Hanna may
be wrong.



           Best,



           Harvey


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