The Collatz conjecture algorithm applied to negative integers

No, see the wikipedia article. If we apply the Collatz function on any integer, then it is conjectured that it ends up in one of five* "cycles":

• $ \ldots \to 1 \to 4 \to 2 \to 1 \to \ldots$
• $\ldots \to −1 \to −2 \to −1 \to \ldots$
• $\ldots \to −5 \to −14 \to −7 \to −20 \to −10 \to −5 \to \ldots$
• $\ldots \to −17 \to −50 \to −25 \to −74 \to −37 \to −110 \to −55 \to −164 \to −82 \to −41 \to −122 \to −61 \to −182 \to −91 \to −272 \to −136 \to −68 \to −34 \to −17\to \ldots$
• $\ldots \to 0 \to 0 \to 0 \to \ldots$ (note that you can only end up in this cycle if you started with $0$)

(* Thanks to @user144527 for mentioning the trivial cycle $0 \to 0 \to \ldots$)

Applying the map 1+3x for x odd, x/2 for x even, to negative integers, has the same result as applying the map 3x-1 for x odd, x/2 for x even, to positive integers (except for the sign of the result). This is why you find that the map 1+3x on negative integers has cycles other than the trivial one. These cycles are mirror images of 3x-1 cycles on positive integers such as {5,7,5} and {17,25,37,55,41,61,91,17} (showing odd values only here). There may be more non-trivial cycles of the 3x-1 map on positive integers.