Triple intersection numbers of Q-polynomial distance-regular graphs

We use the system of linear Diophantine equations introduced by Coolsaet and Jurišić to obtain information about a feasible family of distance-regular graphs with vanishing Krein parameter q221 and intersection arrays {(r+1)(r3−1),r(r−1)(r2+r−1),r2−1;1,r(r+1),(r2−1)(r2+r−1)},r≥2. In this way we are able to calculate certain triple intersection numbers and prove nonexistence for all r≥3. For r=3 nonexistence was not known before, however it is well known that the intersection array for r=2 uniquely determines the halved 7-cube. Then we show how to apply Terwilliger balanced set conditions for Q-polynomial distance-regular graphs to produce additional linear Diophantine equations.