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, 3 months ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$

Sort by:

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, 3 months ago

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

- 4 years, 2 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, 2 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, 2 months ago

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

- 4 years, 3 months ago