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[a1s]

is the pattern "each number (after the first two) is the sum of the previous two"?

[Naturegirl1999]

is the pattern "each number (after the first two) is the sum of the previous two"?

Close enough, yeah
“ x+previous number, if no previous, x adds itself, then starts adding the previous number”

Looking back, I might have messed something up in validity checks

Who wants to go?

[coalboat]

Pattern:

Spoiler (click to show/hide)

if a(n) ends in 0, a(n+1) = 1

if a(n) ends in 1, a(n+1) = a(n) - 1

if a(n) ends in 2, 3, 4, a(n+1) = a(n)^2 - 2

if a(n) ends in 5, 6, 7, 8, 9, a(n+1) = a(n)^3 - 3

Example:
3, 7, 340, 1, 0, 1, 0, 1, 0...

[Naturegirl1999]

4,8,440,1,0,1,0?

[coalboat]

4,8,440,1,0,1,0?

No

[Naturegirl1999]

0,1,2,1,0,1?
1,0,1,0,1,0?
1,1,1,0,0,0?

[coalboat]

0,1,2,1,0,1?
1,0,1,0,1,0?
1,1,1,0,0,0?

no
yes
no

[a1s]

0,1,0,1,0?
4,8,450,1,0,1,0?
1,1,1,1,0,1,0?
1,1,2,1,0,1,0?
1,1,3,1,0,1,0?
1,1,4,1,0,1,0?
1,1,5,1,0,1,0?
1,1,6,1,0,1,0?
1,1,7,1,0,1,0?
1,1,8,1,0,1,0?
1,1,9,1,0,1,0?
1,1,10,1,0,1,0?

[coalboat]

yes:
0, 1, 0, 1, 0

no:
4,8,450,1,0,1,0
1,1,1,1,0,1,0
1,1,2,1,0,1,0
1,1,3,1,0,1,0
1,1,4,1,0,1,0
1,1,5,1,0,1,0
1,1,6,1,0,1,0
1,1,7,1,0,1,0
1,1,8,1,0,1,0
1,1,9,1,0,1,0
1,1,10,1,0,1,0

[a1s]

are any of these good?
340,1,0,1,0,1
1,1,11
1,1,12
1,1,13
1,1,14
1,1,15
1,1,16
1,1,17
1,1,18
1,1,19
1,1,20

[coalboat]

yes:
340,1,0,1,0,1

no:
1,1,11
1,1,12
1,1,13
1,1,14
1,1,15
1,1,16
1,1,17
1,1,18
1,1,19
1,1,20

[coalboat]

I offer two more examples:

2, 2, 2, ...

1000, 1, 0, 1, 0, ...

[a1s]

33, 7, 340, 1?
17,340,1,0?
10,1,0,1?

[coalboat]

yes:
10,1,0,1

no:
33, 7, 340, 1
17,340,1,0

[a1s]

20,1,0,1
30,1,0,1

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
4,14
4,15
4,16
4,17
4,18
4,19
4,20