Recurrence relation $T_{k+1} = 2T_k + 2$ I have a series of number in binary system as following:
0, 10, 110, 1110, 11110, 111110, 1111110, 11111110, ...

I want to understand : Is there a general seri for my series?
I found this series has a formula as following:
(Number * 2) + 2
but i don't know this formula is correct or is there a general series (such as fibonacci) for my issue.

$T_{k+1} = 2T_k + 2$. Adding $2$ to both sides, we get that $$\left(T_{k+1}+2 \right) = 2 T_k + 4 = 2 \left( T_k + 2\right)$$
Calling $T_k+2 = u_k$, we get that $u_{k+1} = 2u_k$. Hence, $u_{k+1} = 2^{k+1}u_0$. This gives us $$\left(T_{k}+2 \right) = 2^k \left( T_0 + 2\right) \implies T_k = 2^{k+1} - 2 +2^kT_0$$
Since, $T_0 = 0$, we get that $$T_k = 2^{k+1} - 2$$ where my index starts from $0$.
