Pell numbers

[Creighton Dement]

Dear Seqfans, 

Here is a relation I came across involving Pell numbers which reminds me
of a conversation I had with a seqfan a few days ago...

The conversation had to do with sequences of the form 
c(2n) = a(n), c(2n+1) = a(n) (or c(2n+1) = -a(n)) and that, apparently,
such sequences may often be used in relating two or more sequences. See,
for ex. my comment for A061547 or A100434 (this case is very similar to
the one at hand as it also involves Pell numbers) .

Define: (a(n)) =  (0, -1, -1, -8, -8, -49, -49, -288, -288, -1681,)
http://www.research.att.com/projects/OEIS?Anum=A001108
a(n)-th triangular number is a square: a(n+1) = 6*a(n)-a(n-1)+2.

Define (b(n)) =  (0, 0, 3, 3, 20, 20, 119, 119, 696, 696, 4059, 4059,)
 http://www.research.att.com/projects/OEIS?Anum=A001652
a(n)=6a(n-1)-a(n-2)+2 with a(0) = 0, a(1) = 3.

Conjecture: a(n) + b(n) = ((-1)^n)*A000129(n)

I know this is an abstract question, but does anyone see a reason
(geometric or otherwise) why these sequences choose to "double up on
themselves" ?

Sincerely, 
Creighton