Captain reported a landing gear unsafe indication on take off resulted in a return to departure airport and precautionary landing.

Narrative

Nothing abnormal or anything out of the ordinary until calling for 'Gear down; landing checklist.' As soon as the First Officer finished his flow; he points to the landing gear indications and states that the right main gear isn't indicating locked; and still showing unsafe with the gear doors also open. He suggests a missed approach. I agree and we tell ZZZ Approach that we need to execute a missed approach for a landing gear unsafe indication. I was hand flying at the time; and starting down on glide path just under 4;000 feet MSL. Approach tells us to fly present heading and climb to 4;000 feet. I add power to climb; the First Officer sets the flight guidance; we take note of maintaining 215 or less; and the airspeed stays just under 200 knots. As we get vectored to downwind I [request priority handling] with ATC and we begin to run the alternate gear extension checklist. XX souls; about 1:20 minutes of fuel on board. Through conversation earlier on the trip; we had talked coincidentally about how long the alternate gear extension checklist takes; and to not dilly dally. I call the flight attendants and ask about what they can see with the landing gear; they say that they see them as both down and looking like they are in position. I make a PA to the passengers that we have a landing gear indicator light that isn't showing correctly and that we are going to run checklists to fix it. The First Officer ended up getting out of his seat to pump the main gear as the light didn't illuminate; and the alternate indications in the floor also showed that the right main was not secure. The alternate gear extension checklist did not fix the problem; as the First Officer moved onto the alternate gear extension failure checklist; I get on SATCOM and contact Maintenance Control. They tell us to run the associated checklists and ask our intentions; and then transfer us to Dispatch. We relay the events and what we had done; and I state my intentions again; to run the checklists as much as possible; then return to landing before we run out of gas no matter the outcome.At this point; ATC has us holding north of ZZZ1 at the ZZZZZ fix. There's a 50 knot wind out of the south; and I take note as to when we need to head towards the field to have fuel to land with. The alternate extension failure checklist does not produce results. We review gear unsafe landing checklist; then proceed to re-run the alternate gear extension checklist as driven by the failure checklist. No result. At this point we have about 1;800 pounds of fuel remaining; are north of ZZZ1 with a headwind; and I determine it's time to make for the airport. I call the flight attendants and advise them that we will be shutting down the #2 engine; and give them the [preparation] report for cabin prep; and brief our bracing signal of emergency lights on and brace brace brace over the PA when we get to about 500 feet. I then make a PA to the passengers about shutting down the engines and precautions of trucks on the runway; to follow Flight Attendant instructions; and thank them for their compliance. We review the gear unsafe landing checklist again; then go to the [specific] checklist to shut down the #2 engine in the event that the gear does collapse on landing. We shut down the engine; run the engine clean up items checklist; brief Runway XXL ILS; landing flaps 35 planning to land on the left side of the runway..and we proceed inbound. We configure early; the conditions were nice for a single engine approach; wind about 10 knots no ice; no turbulence; at 500 AFE turn on the emergency lights and give the brace signal over the PA and land. Come to a stop on the runway; and run the checklist to shut down the number 1 [engine]. We talk with Tower and Airport Ops about staying on the runway; and I try to figure out what's next. We have a tow team coming; but I ask about getting buses. Remembering the gear gets pinned during a tow; I feel confident about leaving passengers on board during the tow. I make a PA to the passengers about remaining in their seats and thanking them for their compliance. I step out of the plane to go talk with the tow team and notice that the gear doors are still open from the alternate gear extension checklist. I call Maintenance Control and they say to start an engine to energize the hydraulics to shut the door. I coordinate with the tow team and Airport Ops about starting the engine; then we will be towed to the gate. Getting back to the flight deck we secured all the doors; ran the originating and before start checklists. We attempted to start #1 and got no light off. Ran the engine start abort checklist. Tried to start #2; no light off. Ran abort checklist. I wasn't having a good day at this point; and when I had called Maintenance Control had noticed a text from Name; so I called Name for some grounding and guidance. As I said hello; I noticed both T handles were still pulled. I laughed to myself; and put them back in. We started the number 1 with no issue; energized the hydraulic systems; shut down the #1 and got towed to the gate. I had made announcements to the passengers before and during this process. At the gate; we deplaned normally. A Flight Attendant Supervisor was waiting; and talked with the flight attendants and removed them from flying for the rest of their day. Crew Scheduling tried to assign a ZZZ2 turn to the First Officer and he said he was fatigued. (I don't think they knew what just happened) then I got our crew together and debriefed a little bit. Talked about what we experienced; how we felt; and that we were all done for the day. Afterwords; I called Name again and told him the story that I'm writing now; and we talked about some of the choices and decisions that our crew had made. I felt like my First Officer; flight attendants and passengers had all done a remarkable job with the situation dealt to us. The mechanic who met us at the airplane had stated that the plane had recently undergone some extensive landing gear work; so maybe that was a part of the faulty indications. I'm grateful the gear worked as desired and I feel that shutting down the engine gave me a lot of confidence proceeding to the landing. Same as aboveNext time I would restart all flows and reconfigure the airplane to a starting position after setting everything up for the engine start. The T handle situation I wish I had noticed; but I was parked on [Runway] XXL; had just had a rather stressful event; and we caught it eventually.I think a Sim Instructor would have done a good job resetting the plane; but it was my first time pulling T handles in the actual plane. I was very impressed how well the plane flew single engine; it really was a low stress aspect to the event. I also felt that choosing to shut down the engine mitigated a very large threat and danger to the passengers; so I stand by that decision.