1. There's no indication that anything preceded the first -1.

@LovinLarge you are dialectically correct (it was inadvertently remiss of me to imply that) however the domain of this set of integers has been deliberately constrained to start at that point to conceal earlier members of the set (range).

@waitingforgodo To be accurate, I think you had a duty to disclose that earlier members of the set existed, and you could have done so while still concealing what those members were, like this: ... -1, 3, 1, -1, 3 ....

@LovinLarge in riposte, there was no duty of disclosure: due diligence (and proper analysis) would have revealed the implicit truncation. The truncation of the series does not alter the correct answer which is obtained using finite differences to derive the formula.

Harrumph! You ASSume that the information given is complete. (To ASSume, to make an ass out of you and me).

Prove me wrong.

More seriously, the sequence 7,10,17,7,111,26,34,16,24,32,107,14,14,22 ... 41,41,134,134,41,41 ...arises naturally from observations on the Collatz Conjecture.

@anglophone You failed to prove me wrong. I was given a very simple repeating sequence so I repeated the sequence, more sequence would have informed me of the relationship to the Collatz Conjecture, but you didn't provide an adequate hint, besides no one has been able to solve the Collatz Conjecture.
If the previous term is even, the next term is one half of the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.
So not true to the Collatz Conjecture to begin with.
The Collatz Conjecture has to start with any positive integer n and -1 isn't a positive integer.
There's no even numbers in your sequence.
So I used the kiss principle being a good engineer, since the best designs are always the simplest.

@Willow_Wisp I find it odd that you should say that there are no even numbers in my sequence. Perhaps ternary sequences are more to your taste?

Oh, of kiss: "Keep It Simple, Sweetheart".

And now I am about to be exceedingly rude: the simplest design of the Tay Rail Bridge did not seem to work so well: [en.wikipedia.org] (On a personal note, I once lived not far from the location of that sad event.)

@anglophone I think you're just pulling numbers out of your ass frankly and thinking any number in your sequence is even tells me you can't count much less do math. My job is in part about programming serial communications devices trust me I know strings of zeros and ones. Add a 2 into the mix and it's ternary. The best designs are simple, not stupid, they just didn't understand harmonics back then so situations like the Tacoma Narrows Bridge happened. The Tay Rail Bridge was a British engineering feet, congrats on not thinking the wind was a thing.

@Willow_Wisp You want the algorithm? Easy! Take any number of the form 3+6n, n being a non-negative integer, and count the number of iterations of the Collatz function to arrive at 1. As the lighthouse said to the warship: "Your move!".

@anglophone You started your sequence with a negative 1 then called it the Collatz Conjecture a sequence which explicitly start with a POSITIVE INTEGER ONLY. So I abandon the question like a defective Fiat, which stands for fucking inadequate transportation.

@Willow_Wisp Um, when did negative 1 equal 7? Are you speaking octal?

Fiat? Hold En? De Troit? Trouvé's tricycle?

@anglophone Your poll is, and I quote.

What is the next number in this series?

-1, 3, 1, -1, 3, ...

There's no 7 in the question.

@Willow_Wisp Sorry, not my poll, not my monkeys, not my circus. Perhaps @waitingforgodo would like to add his 31 cents' worth to the discussion.

@anglophone Hah! Right you're not even the person that did this dumb ass poll.... LOL

@Willow_Wisp I recall the Tacoma Narrows bridge and was familiar with the harmonics that brought it down. I got into some exotic higher math in college for electronics and computing but it's all forgotten now. I do know the difference between feet and feat though, fwiw, and have fond memories of a Fiat Spider I had fun driving into the ground, but I agree much of European car engineering wasn't up to par, like a British Leyland Triumph TR6 I was also fond of. I do wish I could follow the rest of the conversation and admit I'm a dufus as far as that goes. Carry on.

solution and methodology posted above.

Not this time, sweetie.

@MsKathleen Every time. The question doesn't matter.