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:
$$a_{3n} = 12n + 2$$ $$a_{9n+1} = 36n + 7$$ $$a_{27n+4} = 108n + 18$$ $$a_{27n+16} = 108n + 66.$$