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[coalboat]

yes:
20,1,0,1
30,1,0,1
4,14

no:
4,1
4,2
4,3
4,4
4,5
4,6
4,7
4,8
4,9
4,10
4,11
4,12
4,13
4,15
4,16
4,17
4,18
4,19
4,20

[a1s]

Spoiler: accepted sequences (click to show/hide)

3, 7, 340, 1, 0, 1, 0, 1, 0
1,0,1,0,1,0?
0, 1, 0, 1, 0
340,1,0,1,0,1
2, 2, 2
1000, 1, 0, 1, 0,
10,1,0,1
20,1,0,1
30,1,0,1
4,14

14, 14
14, 0
14, 1

[coalboat]

no:
14, 14
14, 0
14, 1

The sequence started by 14 is: 14, 194, 37634, 1416317954, ...

[a1s]

Just to be sure, since I haven't asked, these are both valid?

7, 340, 1, 0, 1
4, 14, 194, 37634

[coalboat]

yes:
7, 340, 1, 0, 1
4, 14, 194, 37634

[a1s]

I gave up. I looked at the pattern and it's really complicated. I'm not saying no one could have ever guessed it, but I couldn't.

[coalboat]

I offer two other examples:
9, 726, 382657173, 146426512048351927, 3139490347742890243660109406183545660322742027034980, 1, 0, 1, 0, ...
22, 482, 232322, 53973511682, 2913139963286990469122, ...

The pattern is indeed a bit complicated, but it has a somewhat natural "rhythm" to it.

There's a big number calculator that can help:
https://www.calculator.net/big-number-calculator.html

[a1s]

Welp, I guess this is dead.

Spoiler: here's a new pattern (click to show/hide)

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

2, 4, 6 fits the pattern.

[coalboat]

a(n+1) = a(n) + 2?

[a1s]

a(n+1) = a(n) + 2?

No,
because 10,11,12 also fits the pattern.

[coalboat]

10, 11, 12, 13?

[a1s]

10, 11, 12, 13?

does not fit the pattern.

[coalboat]

10, 20, 30?

[a1s]

10, 20, 30?

yes

[coalboat]

a1 + a3 = 2*a2?