(TLDR: I independently discovered a property of Fibonacci-like sequences and I couldn't have been the first one to notice. Who else has?)
For some background, I haven't been in a math class since high-school (I'm 24) and the highest level class I was in was Algebra 2. This is all just for fun.
I was messing around with some Fibonacci numbers earlier. I was using a generalized version of the sequence that allowed you to more or less start the sequence at any position given two sequential Fibonacci numbers ($x$ and $y$). I was presented it with only the first 4 terms and I expanded it to the 7th:
$$x, y, (x+y), (x+2y), (2x+3y), (3x+5y), (5x+8y)$$
I noticed the Fibonacci sequence poking it's head out in the multipliers of $x$ and $y$ and was able to come up with a generalization, with $F'$ representing the Fibonacci sequence starting at a different position.
$$F'_{n} = F_{n-2}\centerdot x + F_{n-1}\centerdot y$$
However, the generalized sequence mentioned earlier doesn't stipulate $F_{1} = 1$ and $F_{2} = 1$. Because of this, the sequence and equation also apply to Lucas Numbers and any other related sequence, while still retaining the relationship to the Fibonacci Sequence. For this purpose, $F'$ becomes G. Example:
$$G_{n} = F_{n-2}\centerdot x + F_{n-1}\centerdot y\\ G_{1} = F_{-1}\centerdot x + F_{0}\centerdot y\\ G_{2} = F_{0}\centerdot x + F_{1}\centerdot y\\ G_{3} = F_{1}\centerdot x + F_{2}\centerdot y\\ G_{4} = F_{2}\centerdot x + F_{3}\centerdot y\\ G_{5} = F_{3}\centerdot x + F_{4}\centerdot y$$
Applying this to Fibonacci and Lucas:
$$F_{1} = 1\times 1 + 0\times 1 = 1\\ F_{2} = 0\times 1 + 1\times 1 = 1\\ F_{3} = 1\times 1 + 1\times 1 = 2\\ F_{4} =1\times 1 + 2\times 1 = 3\\ F_{5} = 2\times 1 +3\times 1 = 5$$
$$L_{1} = 1\times 1 + 0\times 3 = 1\\ L_{2} = 0\times 1 + 1\times 3 = 3\\ L_{3} = 1\times 1 + 1\times 3 = 4\\ L_{4} =1\times 1 + 2\times 3 = 7\\ L_{5} = 2\times 1 +3\times 3 = 11$$
This essentially allows you to calculate any term in any Fibonacci-like sequence. For example: If $G_{1} = 9$ and $G_{2} = 4$, then $G_{11} = 526$ (I double-checked the sequence by hand) $$G'_{11} = F_{9}\centerdot 9 + F_{10}\centerdot 4 = 526$$
My question is this: Is there a name for this generalization? I can't seem to find anything on it and Google isn't very good at searching equations. This seems so simple that there HAS to be something out there about it; I couldn't have discovered it. Is there any documentation of this?
(Side note: I also can't seem to find the collective term for these sequences, which is why I've been referring to them as Fibonacci-like.)