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What about the sequence \(2, 4, 10, 28, 82, \ldots\text{?}\) Here the terms are always 1 more than powers of 3. That is, we have added the sequences \(1,1,1,1,\ldots\) and \(1,3,9, 27,\ldots\) term by term. Therefore we can get a generating function by adding the respective generating functions:

\begin{align*} 2 + 4x + 10x^2 + 28x^3 + \cdots \amp = (1 + 1) + (1 + 3)x + (1 + 9)x^2 + (1 + 27)x^3 + \cdots\\ \amp = 1 + x + x^2 + x^3 + \cdots + 1 + 3x + 9x^2 + 27x^3 + \cdots\\ \amp = \frac{1}{1-x} + \frac{1}{1-3x} \end{align*}

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