Simple-looking sequences $A$ and $B$ defined by a complementary equation Define $A=(a_n)$ and $B=(b_n)$ by $b_0=1$ and 
$$a_n=b_n+b_{2n}$$
for  $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$.  Can someone prove that
$$b_{3n+2}=4n+4$$
for $n \geq 0$?  Initial terms:
$$A=(2,7,10,14,18,23,26,31,34,38,43,46,50,\ldots)$$
$$B=(1,3,4,5,6,8,9,11,12,13,15,16,17,19,20,\ldots).$$
This question resembles Limit associated with complementary sequences, but I don't see how the method of solution there applies here.  The sequences $A$ and $B$ are https://oeis.org/A304799 and https://oeis.org/A304800. 
 A: This is another observation showing that, after all, the pattern is only partially regular. The diagram refers to Greg Martin's comment with the observation that for $n\equiv1\pmod3$, the choice between $b_n=\lfloor 4n/3+7/6 \rfloor$ and $b_n=\lceil 4n/3+7/6 \rceil$ does not follow an easy pattern. It appears that rounding down happens more often than rounding up, and it is the excess which I have displayed here.

Reading example: for the $500$ such values below $n=1500$, rounding down happens $102$ times more often than rounding up (i.e. $301$ times down versus $199$ times up).
While the excess seems essentially linear (red line, slope about $1/15$, thus $60$% rounding down and $40$% rounding up), the bumps and dents are quite irregular.
Not that this helps for the original question, but it gives more evidence for the fact that the sequences $A$ and $B$ are not that simple-looking as it may seem!
A: This is only an observation and not a solution, but perhaps it may save some time for anyone else who looks at this problem. So far as the sequences are given in OEIS, the following formulae (the second of which is given by the proposer) fit the data:
$$b_{3n} = 4n + 1$$
$$b_{3n+2} = 4n + 4$$
$$b_{9n+1} = 12n + 3$$
$$b_{18n+7} = 24n + 10$$
$$b_{27n + 4} = 36n + 6$$
$$b_{27n + 16} = 36n + 22.$$
If those formulae are correct, then the original condition that $a_n = b_n + b_{2n}$ would give determinations of $A$ in the following cases:
$$a_{3n} = 12n + 2$$
$$a_{9n+1} = 36n + 7$$
$$a_{18n+7} = 72n + 31$$
$$a_{27n+4} = 108n + 18$$
$$a_{27n+16} = 108n + 66.$$
(Formula for $a_{18n+7}$ corrected Aug 29 2019.)
