I have the following sequences:
$a = [0, 1, 2, 3, 4, 5]\\ b = [1, 2, 4, 7, 13, 26]\\ c = [1, 1, 2, 4, 7, 13]$
I'm trying to reverse-engineer an undocumented API and believe that the values in $a$ can be used to determine the corresponding values in $b$ and $c$. For instance, we could say that third element in $b$ ($4$) is two times the value in $a$ ($2$), while the corresponding value in $c$ is the same as $a$'s. Unfortunately that algorithm doesn't work across all the values of the sequences. Does this make sense? If so, can anyone think of any common series that would give the values of $b$ and $c$$`$?