# Sequences

I have an interesting sequence whose properties I do not know yet(that is why I'm asking for your help). Define $$a_{1} = 1$$ and $$a_{2} = 1$$ for this sequence. From then on, $$a_{n}$$ will be defined as $$a_{n-2} + a_{n-1}$$ if $$n \equiv 3 \mod {4}$$, $$a_{n-2} - a_{n-1}$$ if $$n \equiv 0 \mod {4}$$, $$a_{n-2} \times a_{n-1}$$ if $$n \equiv 1 \mod {4}$$, and $$a_{n-2} \div a_{n-1}$$ if $$n \equiv 2 \mod {4}$$. The first few terms of the sequence are $$1, 1, 2, -1, -2, \frac {1}{2}, -\frac {3}{2}, ...$$. I haven't found much, but I've hypothesized that similar sequences(mixing up the order of adding, subtracting, multiplying, and dividing) will result in a finite sequence(stopping when a term is undefined) if and only if a 0 appears(the iff is important). Anyone want to find something about this?

Note by Tristan Shin
4 years, 6 months ago

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Well, the sequence starting from $$1,0,\ldots$$ defies your conjecture.

It is a recursive sequence with repeated segment $$[1,0,1,-1,-1]$$

- 4 years, 6 months ago

@Daniel Liu Your sequence has A2 as 0 but it is supposed to be 1

- 4 years, 5 months ago

I don't see why $$a_2$$ can't be $$0$$. The OP said $$a_1=a_2=1$$, but I can surely change that, can I? Or can I only mix up the definitions for how to find later terms?

- 4 years, 5 months ago

I meant only definitions for later terms. The first two terms must stay at 1, otherwise there are several contradictions(including yours).

- 4 years, 5 months ago

The sequence is defined with $$a_2$$ as $$1$$, but your sequence shows $$a_2$$ as $$0$$.

- 4 years, 6 months ago