Problem #PRU-107992

Problem

For each pair of real numbers $a$ and $b$, consider the sequence of numbers $p_n=\lfloor 2\{an+b\}\rfloor$. Any $k$ consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length $k$ is a word of the sequence given by some $a$ and $b$ for $k=4$; when $k=5$?

Note: $\lfloor c\rfloor$ is the integer part, $\{c\}$ is the fractional part of the number $c$.