As the title suggests, the
- start $a_n$,
- end $b_n$,
- height $h_n$
of $n^{th}$ stair in a stair-like integer sequence $F_n$ is known, where
- $F_n=h_i$, iff $a_i\le n\le b_i$, and
- $F_n$ changes linearly from $h_i$ to $h_{i+1}$, if $b_i\lt n\lt a_{i+1}$.
For example, if
- $a_n=3\times 2^{n-1}=\{3,6,12,24,48,\cdots\}$,
- $b_n=\lceil{9\times 2^{n-2}}\rceil=\{5,9,18,36,72,\cdots\}$,
- $h_n=\lceil{3\times 2^{n-2}}\rceil=\{2, 3, 6, 12, 24,\cdots\}$,
then
- $F_n=0,1,2$ for $n=1,2,3$,
- $F_n=h_1=2$ for $a_1=3\le n\le 5=b_1$,
- $F_n=2,3$ for $n=5,6$,
- $F_n=h_2=3$ for $a_2=6\le n\le 9=b_2$,
- $F_n=3,4,5,6$ for $n=9,10,11,12$,
- ...
My attempt: The sequence in this example can be expressed in a recursive manner.
$$ F_n=F_{n-1}+[\exists i \quad\text{s.t.}\quad b_i\lt n\le a_{i+1}], $$
where $[P]$ is the boolean function.
(Actually I'm mainly interested in this example, but I also wonder if this can be generalized!)