B737 Captain reported an unusual amount of noise from the nose wheel area after gear retraction. The flight crew continued to destination airport and after landing; the Captain observed more wear on the right nose tire which caused a slight nose wheel shimmy while taxiing to the gate.

Narrative

After taking off from ZZZ2 on a warm day and high ground speed we heard two metallic sounding 'clunks' as the nose gear came up and then just the normal sound of the snubber. The aircraft flew normally and the engine and hydraulic indications were all normal. Noise levels were also normal so we were satisfied that the gear doors were closed properly. After flying the 73 for more than 30 years I had never heard the gear make that noise as it retracted and I was concerned that there may be some kind of damage to a tire or the linkage. After discussing it with the First Officer (FO) I decided that it would be prudent to consult dispatch and Maintenance Control and let them know my concerns. I asked dispatch to let ZZZ1 know that we would be [requesting priority handling] and asked them to have emergency vehicles ready when we landed. FO and I discussed possible scenarios and reviewed gear problems in the QRH. About an hour from landing I briefed the lead Flight Attendant (FA) of the issue and gave her the briefing and informed her that she I did not need to prepare for an evacuation. I informed the passengers of the situation as we started our descent. We [requested priority handling] when we got to Center and we were cleared directly to the airport. We extended the gear a bit earlier than normal to make sure that it would extend normally and the landing preceded normally. We taxied clear of the runway and came to a stop to let emergency services inspect the gear and then taxied to the gate. There was a slight shimmy in the nose gear after landing but nothing excessive. After reaching the gate and inspecting the gear the only abnormality that I could find was a greater amount of wear on the right tire than the left.