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[delphonso]

1, 3, 4, 3, 1
1, 3, 2, 5, 7
?

[a1s]

1, 3, 4, 3, 1
1, 3, 2, 5, 7
?

yes to both. Would you like to guess the pattern?

[delphonso]

It's alternating or mirroring odds and evens, right?

[a1s]

It's alternating or mirroring odds and evens, right?

Can you be more specific? what is your guess? (feel free to post both if you have 2)

[delphonso]

I'm going with mirrored odds and evens.
1, 2 (mirror) 2, 1
Or
1, 2, (mirror) 3, 4

[a1s]

I'm going with mirrored odds and evens.
1, 2 (mirror) 2, 1
Or
1, 2, (mirror) 3, 4

You got the right of it. Even though 1,2,3,4 clearly isn't mirrored, so I'm guessing it's a typo. 

number parity (even or odd) must be a palindrome (the same backwards as it is forwards).
So 4-7-6 (even-odd-even) is in
and 4-7-8-6 (even-odd-even-even) is not

Would you like to think of the next pattern?

[delphonso]

Haha - Christ. Yeah, 1243 is what I meant to put... That was cool - I think it was whoever posted like 25 1's that got me thinking it didn't have anything to do with math.

Sure, I have an idea.

Spoiler (click to show/hide)

Numbers alternate between starting with a vowel or consonant. Patterns can start with either a vowel-leading or a consonant-leading number.
Ex: 'e'leven, 't'en, 'o'ne. (Letter doesn't matter, just vowel or consonant.)

1, 2, 8, 10 fits the pattern

[a1s]

1, 2, 3, 4?

[coalboat]

number parity (even or odd) must be a palindrome (the same backwards as it is forwards).
So 4-7-6 (even-odd-even) is in
and 4-7-8-6 (even-odd-even-even) is not

Good one!

As for the new pattern:
2, 4, 16, 20
?

[delphonso]

1, 2, 3, 4?

No.

As for the new pattern:
2, 4, 16, 20
?

No

[a1s]

1, 2, 8?

[delphonso]

1, 2, 8?

Yes

[a1s]

1,1?

[delphonso]

1,1?

No.

[a1s]

2,8?