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.

 

[tacman]

?

It seems like this sequence is defined recursively, where each term is a product of two numbers and the next term is dependent on the previous two terms. The first two terms are given, and then the pattern is established for the rest of the sequence.

I have not personally come across this specific sequence before, but it does remind me of other sequences that involve products of numbers and have a recursive definition. It would be interesting to explore the properties and behavior of this sequence further.