This is more a "recreational" problem. By another question I came to the question for a closed-form-formula for this sequence $\small 1 , 2 , 3 , 6 , 9 , 18 , 27, \ldots $ which is just the mixture of the sequences $\small 3^k $ and $\small 2 \cdot 3^k $ .
I tried to find the "Binet"-type expression for it (like for instance for the Fibonacci-sequence) but do not find the initial "key". What is the way to such a formula?