# 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.

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The series is...

$T_k = 2^k - 2$

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thanks for your answer. Do have the seri any name? – MJM Jun 12 '12 at 15:17
no i figured it out because $2^k-1$ looks like all 1s, ie 111111 is a power of 2 less 1. So a number like 111110 is 1 less than this, so $2^k -2$ – Andrew Tomazos Jun 12 '12 at 15:18
$2^k-2$ is so short and succinct that the sequence does not need any shorter name. – Henning Makholm Jun 12 '12 at 15:22

$T_{k+1} = 2T_k + 2$. 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 $0$.

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+1 for derivation of the closed form. – Shaktal Jun 12 '12 at 15:45