Output a unique sign sequence
05AB1E, 14 bytes
∞v®yÒgm=Ox&lt;.±,

Try it online!
My first non-trivial 05AB1E answer! Happy for suggestions to improve it.
The code prints two interleaved sign sequences, both related to the Pólya conjecture. In 1919, George Pólya conjectured that the majority (no less than half) of positive integers up to any finite limit \$\ge2\$ have an odd number of prime factors, counted with multiplicity. Mathematically, the summatory Liouville function \$L(n)\$ (A002819) was posited to satisfy the inequality
$$
L(n)=\sum_{k=1}^n\lambda(k)\le0\
$$
for all \$n\ge2\$, wherein
$$
\lambda(k)=(-1)^{\Omega(k)}=\begin{cases}-1,\ \text{$k$ has an odd number of prime factors}\\\phantom{-}1,\ \text{$k$ has an even number of prime factors}\end{cases}
$$
is the Liouville function (A008836), related to the parity of the number of prime factors of \$k\$, and \$\Omega(k)\$ is the prime-factor-counting omega function that respects multiplicity.
The Pólya conjecture was shown to be false in 1958 when Brian Haselgrove proved the existence of an enormous counterexample near \$n=\exp(831.847)\approx1.845\times10^{361}\$. Subsequently, it has been shown that the smallest counterexample is \$n=906\,150\,257\$, and that the conjecture fails for most values of \$n\$ in the range \$906\,150\,257\le n \le 906\,488\,079\$. It is also now known that \$L(n)\$ changes sign infinitely often.
The code prints two values for each integer \$n\ge1\$:
$$
\DeclareMathOperator{\sgn}{sgn}
\begin{gather}
\lambda(n), \tag{1} \\
\sgn\bigl[2L(n)-1\bigr]. \tag{2}
\end{gather}
$$
The first value is the \$n\$th term of A008836. The second value, chosen to avoid zeros in \$L(n)\$, is (arguably) more interesting. Given the results quoted above, we see that sequence \$(2)\$ alternates between \$-1\$ and \$1\$ infinitely often. However, of its first \$906\,150\,257\$ values, only two (the first and the last) are \$1\$; the intervening \$906\,150\,255\$ values are all \$-1\$.
Commented code
∞v             # iterate over all positive integers y
  ®            # push -1
   yÒ          # push list of prime factors of y (with duplicates)
     g         # length of this list; yields Ω(y)
      m        # exponentiate top two stack items; yields -1**Ω(y) = λ(y)
       =       # print λ(y), keeping it on the stack
        O      # sum the stack; yields L(y)
         x&lt;    # push 2*L(y)-1, keeping L(y) on the stack
           .±, # print the sign