To solve the problem, we analyze sequences arranged in a circle where each number equals the sum of its two neighbors.

Key Observations:

For a sequence (a_1, a_2, ..., a_n) in a circle ((a_0=an, a{n+1}=a_1)), the condition is (ai = a{i-1} + a_{i+1}) for all (i).

• n=3: All terms must be zero (trivial solution).
• n=4: All terms must be zero (trivial solution).
• n=5: All terms must be zero (trivial solution).
• n=6: Non-trivial sequences exist. For example: (1,0,-1,-1,0,1) (each term is the sum of its neighbors).

Conclusion:

The smallest (n>2) with non-trivial solutions is 6.

Answer: (\boxed{6})

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