Balancing Diophantine triples with distance 1

For a positive real number w let the Balancing distance ‖w‖B be the distance from w to the closest Balancing number. The Balancing sequence is defined by the initial values B0 = 0, B1 = 1 and by the binary recurrence relation Bn+2 = 6Bn+1 − Bn , n ≥ 0. In this paper, we show that there exist only one positive integer triple (a, b, c) such that the Balancing distances ‖ab‖B , ‖ac‖B and ‖bc‖B all are exactly 1.