What was found
TAKEOFF ROLL WAS UNEVENTFUL. DURING THE ROTATION, FELT A SHAKING IN THE LEFT REAR OF THE AIRCRAFT IMMEDIATELY BEFORE LIFTOFF. THE SHAKING STOPPED ONCE AIRBORNE. FELT LIKE THE AIRCRAFT WAS SKIPPING TO THE SIDE DUE TO A LITTLE SIDE LOAD FROM THE CROSSWIND. THE FLIGHT WAS UNEVENTFUL UNTIL LOWERING THE GEAR DURING APPROACH. WITH THE GEAR HANDLE DOWN, SAW THE LEFT MAIN WAS SHOWING A RED UNSAFE LIGHT AND THE GEAR HORN WAS SOUNDING. CYCLED THE GEAR AND THE SAME THING HAPPENED. PERFORMED A GO AROUND TO RUN THE CHECKLIST. ACCOMPLISHED THE GEAR DISAGREE, AND ONE OR BOTH MAIN LANDING GEAR POSITION INDICATORS SHOW UNSAFE CHECKLISTS. WITH THE GEAR HANDLE DOWN, THE MAIN GEAR INDICATORS SHOWED THE RED LEFT MAIN LIGHT, BUT THE SECONDARY MAIN GEAR INDICATOR SHOWED GREEN. RETRACTED THE GEAR AND THE RED LIGHT TURNED OFF, BUT THE SECONDARY GREEN LIGHT STAYED ILLUMINATED. LOWERED THE GEAR AGAIN. PERFORMED THE FLY-BY WITHOUT INCIDENT. THE TOWER SAID THE GEAR APPEARED TO BE DOWN AND LOCKED. DECLARED AN EMERGENCY. TOUCHDOWN WAS SMOOTH, BUT THE STRONG VIBRATION IN THE BACK LEFT HAPPENED ALMOST IMMEDIATELY. NO BRAKING APPLIED. THE TOWER REPORTED SEEING ONE OF OUR LEFT MAIN TIRES BLOWN OUT. AT 70 KNOTS THE VIBRATION DISAPPEARED. SHUT DOWN THE AIRCRAFT ON THE RUNWAY.