Sum of GCD's?!

Algebra Level 4

Consider the sequence $$50 + n^2$$ for positive integer $$n$$:

$51, 54, 59, 66, 75, \ldots$

If we take the greatest common divisor of 2 consecutive terms, we obtain

$3, 1, 1, 3, \ldots$

What is the sum of all distinct elements in the second series?

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