Suppose there are 6 sequences $a=(a_n)_{n\geq 0}, b=(b_n)_{n\geq 0},c=(c_n)_{n\geq 0},d=(d_n)_{n\geq 0},e=(e_n)_{n\geq 0},f=(f_n)_{n\geq 0}$, the data can be seen here: Data. I found out by trial and error that $a$ is defined by \begin{align} a_n=\lceil 970\cdot1.025^{n}\rceil. \end{align} It is easy to see that \begin{align} c_n=\lceil(b_n/2)/5\rceil\cdot5, \\ d_n=\lceil(c_n/3)/5\rceil\cdot5, \\e_n=\lceil(d_n/4)/5\rceil\cdot5, \\ f_n=\lceil(e_n/5)/5\rceil\cdot5. \end{align} I guess $b_n$ should be as well rounded to the next integer divisibly by $5$, but i do not see the explict definition. It should have something to do with $a_n$, or at least with $970$. $b$ is growing on average relativly faster than $a$ until $n\approx 40$ then slower. How can i find $b$ without trial and error?
Would it be helpful to have different examples for the sequences $a$ and $b$? I mean $a'$ with $a'_0=510$ generates a sequence $b'$ with $b'_0=55$ and $a''$ with $a''_0=700$ generates a sequence $b''$ with $b''_0=70$. I can list about the first $30$ members of those sequences as well.
edit: something is not right with my formulas if $f_n=0$, but it should always mean "round to the next integer divisibly by $5$".