Certain product sequences and their factors

[ramsey2879]

define the sequence P_n as follows:

P_{0} = 1 ; P_{1} = a and P_{n} = 6P_{(n-1)}-P_{(n-2)} + 2a^2-8a+4

Then each term is a product of two numbers as follows
P_{n}= {1*1,1*a,a*b,b*c,c*d,d*e,\dots}
where
b = 2a-1
c = 4b-a
d = 2c-b
e = 4d-c
f = 2e-d
...
...

Has anyone come across this before

 

[ramsey2879]

define the sequence P_n as follows:

P_{0} = 1 ; P_{1} = a and P_{n} = 6P_{(n-1)}-P_{(n-2)} + 2a^2-8a+4

Then each term is a product of two numbers as follows
P_{n}= {1*1,1*a,a*b,b*c,c*d,d*e,\dots}
where
b = 2a-1
c = 4b-a
d = 2c-b
e = 4d-c
f = 2e-d
...
...

Has anyone come across this before

Now I generalized this and made a slight adjustment
Let a and b be integers
define the sequence P_n as follows:

P_{0} = ab ; P_{1} = b^{2}-ab and P_{n} = 6P_{(n-1)}-P_{(n-2)} + 2a^{2}-8ab+4b^{2}

Then each term is a product of two numbers as follows
P_{n}= \{a*b,b*c,c*d,d*e,\dots\}
where
c = 4b-a
d = 2c-b
e = 4d-c
f = 2e-d
...
I am thinking of at looking at this further in various finite number systems Z_n but right now I don't know if any of my sequences consist of complete residue sets or if they are just subsets thereof. The constant that is added in the recursive formula can also be written as 4(b-a)^{2}-2a^2 and the sets (c,d) (e,f) (g,h) can also be interchanged for the a and b thereof.