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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!)

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