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?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

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

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

Daniel Liu - 7 years, 1 month ago

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

Nanayaranaraknas Vahdam - 7 years, 1 month ago

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

Mardokay Mosazghi - 7 years, 1 month 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?

Daniel Liu - 7 years, 1 month ago

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

Tristan Shin - 7 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$