Help needed to figure out relationships between a few sequences 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 $c$`?
 A: The sequence $b$ can be obtained from $a$ by the function $$t \mapsto \frac{t^4}{12}-\frac{t^3}{2}+\frac{17 t^2}{12}+1$$ and the sequence $c$ using $$t \mapsto \frac{t^5}{60}-\frac{t^4}{6}+\frac{7 t^3}{12}-\frac{t^2}{3}-\frac{t}{10}+1.$$ I am pretty sure this is not what you want!
A: Unfortunately, we don't have enough information here to reverse-engineer your problem. For instance, the Generalized Catalan Numbers have been referenced in the comments (by Qiaochu Yuan) because they appear in your sequence $b.$ This sequence starts $1,1,1,2,4,7,13,26,52,104,\ldots$
However, I could define a “random” sequence by $u_1=1$ and for $n>1$
$$U_n=2U_{n-1} \quad \textrm{ for } n \equiv 0,1 \textrm{ mod } 4$$
and
$$U_n=2U_{n-1}-1 \quad \textrm{ for } n \equiv 2,3 \textrm{ mod } 4.$$
This follows the stated Catalan sequence exactly, and so contains your sequence $b,$ up until we get 103 instead of 104. I give this as an illustration that with the current information this problem has no definitive answer.
EDIT: To be fair to the question asker, I'd like to point out that the fact that he/she cannot do what he/she wants to do is often not so clear to people who've been brought up on a diet of school mathematics where one is frequently asked to fill in the missing number in a sequence or find the next number.
