An example of a PN sequence for $n=3,\,k=1$ is: $$001,\ 010,\ 101,\ 011,\ 111,\ 110,\ 100.$$
We consider sequences of binary words of length $n$, each containg at least one nonzero bit. Denote $N=2^k-1$. A sequence of distinct words $w_1,w_2,\ldots,w_N$ is said to be a simple PN sequence (PN means "Pseudo Noise") iff there is a number (parameter) $k$ such that for each $i\lt N$ $$w_i=a_0a_1a_2\cdots a_{n-1}\Rightarrow w_{i+1}=a_1a_2a_3\cdots a_{n-1}a_0', \ \mbox{where}\ a_0'=a_0\ \mbox{xor}\ a_k,$$ In other words PN sequence is a (simply generated) Hamiltonian path in a de Bruijn graph containing all nodes except $0^n$. Sequences of this type are also called maximal LFSR sequences (LFSR stands for "Linear Feedback Shift Register").
Observation. If $w_1,w_2,\ldots,w_N$ is a PN sequence then $(n-1)$-length suffix of $w_i$ is not necessarily a prefix of $w_{i+1}$.
In this exercise we discuss how simple PN sequences can be derived from primitive trinomials over $Z_2$ (the field of bits) of the form $P(x)=x^n+x^k+1$. Such a polynomial is primitive if all polynomials $x^i \ \mbox{modulo}\ P(x)$ are distinct, for $1\le i \le 2^n-1$ (it is maximal number of nonzero binary polynomials of degree smaller than $n$).
A very partial list of primitive trinomials is
$x^3+x+1,\ x^3+x^2+1,\ x^4+x+1,\ x^4+x^3+1,\ x^5+x^2+1,$
$x^6+x+1,\ x^7+x+1,\ x^9 + x^4 + 1,\ x^{10} + x^3 + 1,\ x^{100}+x^{37}+1,\ x^{900}+x+1$