A031684 Numbers k such that the continued fraction for sqrt(k) has odd period and if the last term of the periodic part is deleted the two central terms are both 96.

58589, 114293, 187570, 188437, 190177, 283145, 521509, 669413, 675973, 839417, 843085, 1020541, 1024585, 1032697, 1228193, 1239298, 1443026, 1445429, 1447834, 1450241, 1455061, 1459889, 1467146, 1469569, 1949657, 1960841, 1966445