Wire Hive

Let \(a_{n}\) be the number of nodes (red) in the enlarging hive-like structure connecting the nodes in sub-units of equilateral triangles as shown above. For instance, \(a_{1} = 1\); \(a_{2} = 7\); \(a_{3} = 19\); etc.

Starting with \(a_{1} = 1\) as the first perfect square, find the next perfect square value of \(a_{n}\).

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