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There are infinitely many rules. It is easy to find a polynomial of order 5 (or higher) such that the first five numbers are as listed in the question followed by any number that you wish. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

The simplest polynomial of order 4 is:

T(n) = (n^4 - 6n^3 + 15n^2 - 10n + 4)/4 for n = 1, 2, 3, ...

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10y ago
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Q: What is the sequence in 1 3 7 19 51?
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