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That depends entirely on your definition of "valid." If "valid" means "describable by a Turing machine," then the answer is trivially yes. For example, here are three:

• The unique interpolating polynomial of minimal degree.
• The periodic sequence given by repeating what was given.
• The sequence which is constant after what was given.

Edit: Sequence-continuation problems happen to be one of my pet peeves. I do not consider them mathematical exercises, but psychological and cultural: the real goal of any sequence-continuation exercise is to understand what kind of sequences would interest the person who came up with the exercise and what they would consider meaningful. And that is not mathematics.

My favorite example of this is the following sequence: $1, 1, \infty, 5, 86, 3, ...$. How does it continue? (I won't spoil the answer; try looking it up on the OEIS if you're stuck. They write $\infty$ as $-1$.)

2 added 27 characters in body

That depends entirely on your definition of "valid." If "valid" means "describable by a Turing machine," then the answer is trivially yes. For example, here are three:

• The unique interpolating polynomial of minimal degree.
• The periodic sequence given by repeating what was given.
• The sequence which is constant after what was given.

Edit: Sequence-continuation problems happen to be one of my pet peeves. I do not consider them mathematical exercises, but fundamentally psychological and cultural: the real goal of any sequence-continuation exercise is to understand what kind of sequences would interest the person who came up with the exercise and what they would consider meaningful. And that is not mathematics.

My favorite example of this is the following sequence: $1, 1, \infty, 5, 8, 3, ...$. How does it continue? (I won't spoil the answer; try looking it up on the OEIS if you're stuck. They write $\infty$ as $-1$.)

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That depends entirely on your definition of "valid." If "valid" means "describable by a Turing machine," then the answer is trivially yes. For example, here are three:

• The unique interpolating polynomial of minimal degree.
• The periodic sequence given by repeating what was given.
• The sequence which is constant after what was given.

Edit: Sequence-continuation problems happen to be one of my pet peeves. I do not consider them mathematical exercises, but fundamentally psychological: the real goal of any sequence-continuation exercise is to understand what kind of sequences would interest the person who came up with the exercise and what they would consider meaningful.

My favorite example of this is the following sequence: $1, 1, \infty, 5, 8, 3, ...$. How does it continue? (I won't spoil the answer; try looking it up on the OEIS if you're stuck. They write $\infty$ as $-1$.)