If this sequence goes to infinity fast enough then the Gaussian moat-crossing problem is solved and it is impossible to walk to infinity in the complex plane using steps of bounded length stepping only on Gaussian primes.

The first Gaussian prime (restricting ourselves to the first octant) is 1+i which has norm 2 (1^2+1^2). The second is 2+i with norm 5 (2^2+1^2). The difference in those norms is 3, the first term in this sequence.