Can any integer, not a multiple of three, be represented as $n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$? This question has some relevance to the Collatz conjecture. It was originally based on trying to represent a number like this: Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+t_1}3^{n-3} .... + 2^{t_0+t_1+...+t_{n-1}})}{3^n}$ can describe all integers 
However, I generalised that and tried to ask a simpler question instead, now this is simply out of curiosity rather than anything useful, as in often the case with the Collatz conjecture.
Is it possible to represent a number that is not divisible by three as:
$$n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$$
$$n, a, b_i \in \Bbb{Z^+_0}$$
Where $b_i$ is some arbitrary integer such that $b_{i+1} \leq b_i$ 
For example, $13=1+3+9$.
EDIT: My original question asked what happened with these conditions: 
 $$n, a, b_i \in \Bbb{Z^+}$$
$$b_{i+1} < b_i$$
(Notice the subtle differences in specifications.) Under these above conditions, some numbers cannot be represented, like $13$. What are some characteristics of numbers that cannot be represented like this? 
Any help will be much appreciated. Thanks.

This is only a comment, but too long for the comment box 
Since your $b$'s are organized as equal or decreasing one can define "layers" of equal $b$ and sum the terms with equal $2^b$ and decreasing powers of $3$ up. That would give something like
$$ n = 2^{b_1}\left( {3^{A_1} -1 \over 3-1 }\right) + 2^{b_2}\left( {3^{A_2} -1 \over 3-1 } \right) + ... + 2^{b_m}\left( {3^{A_m} -1 \over 3-1 } \right)$$
and can even be expressed more compact like this form
$$ n = 2^{b_1}\left( {3^{A_1} -1 \over 3-1 }+ 2^{b_2}\left( {3^{A_2} -1 \over 3-1 }  + ... + 2^{b_m}\left( {3^{A_m} -1 \over 3-1 } \right)\right)\right) $$
(the indexes $A_k$ and $b_k$ are used only as models, not actual indices here).
From this the question can then also be written in base-3 representation
$$ n=2^{b_1} \cdot 111111...1_3 + 2^{b_2} \cdot 111...1_3 + ...$$
or so. Perhaps looking at such representations would easier give an idea whether any number $n$ can be expressed in such a form. 
