A certain sequence of squares.

[Creighton Dement]

Dear Seqfans, 

Unfortuantely, I have never had the opportunity to see a closed formula
for one of the ("nontrivial") floretion generated sequences of squares
until today. That was perhaps mainly becase I work on the floretion
project exclusively from home and lacked the appropriate software to
investigate deeper until very recently. Anyway, even with the software,
I wouldn't have found the formula were it not for Vladeta Jovovic's
comment, along with the name of the OEIS sequence A025172 
http://www.research.att.com/projects/OEIS?Anum=A025172  
as well as a bit of luck. 


Sequence:  (-1, 1, -25, 49, -1, 529, -1849, 289, -9025, )
Given at http://www.crowdog.de/SeqContext/TriSquare.html
(am currently double checking all formula- hope to submit later today )

It is given by the following formula:

a(n) = (3^(n+1)/2)*(cos((n+1)*arccos(1/3)) + (-1)^(n+1) )

a(n) = a(n-1) + 3*a(n-2) + 27*a(n-3), a(0) = -1, a(1) = 1, a(2) = -25

a(n) = 1/4( p^(n+1) + q^(n+1) ) + (-3)^(n+1)/2  with p = 1 + 2*sqrt(2)i
and
q = 1 - 2*sqrt(2)i  ( i^2 = -1 )

G.f. (1+27*x^2)/((3*x+1)*(9*x^2-2*x+1))


Sincerely, 
Creighton 


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