We consider the greatest common divisor (GCD) of all sums of k consecutive terms of a sequence (S_n)_n≥0 where the terms S_n come from exactly one of following six well-known sequences’ terms: Pell P_n, associated Pell Q_n, balancing B_n, Lucas-balancing C_n, cobalancing b_n, and Lucas-cobalancing c_n numbers. For each of the six GCDs, we provide closed forms dependent on k. Moreover, each of these closed forms can be realized as braid sequences of Pell and associated Pell numbers in an intriguing manner. We end with partial results on GCDs of sums of squared terms and open questions.