What type of sequence is this?

Calculus Level 5

Find the value of \(S\) as defined below:

\[S=\dfrac12+\dfrac2{2^2}+\dfrac3{2^3}+\dfrac{6}{2^4}+\dfrac{11}{2^5}+\dfrac{20}{2^6}+\dfrac{37}{2^7}+\dfrac{68}{2^8}+ \cdots \]

Enter 666 if you come to the conclusion that the series fails to converge.

Note that the numerators follow the pattern \( T_{n+3} = T_{n+2} + T_{n+1} + T_n \).

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