I saw that when the Messenger of Allah (may Allah's peace and blessings be upon him) arrived in Makkah, he would touch the Black Corner in the beginning of his Tawāf and walk briskly for three rounds

The Prophet (may Allah's peace and blessings be upon him) set out to Dhu al-Hulayfah, the Miqāt (place where one assumes Ihrām) for the residents of Madīnah, intending to perform the Farewell Hajj in which he bid farewell to the Sacred House and the rituals of Hajj. He also bid farewell to the people and conveyed his message to them and made them bear witness to that. He assumed Ihrām for both Hajj and ‘Umrah. Thus, he performed Qirān Hajj. Since Qirān Hajj involves Tamattu‘, the people entered into Tamattu‘ along with him. Some people assumed Ihrām for Hajj and ‘Umrah, while others assumed Ihrām for ‘Umrah, intending to perform Hajj next. Some others assumed Ihrām for Hajj only. The Prophet (may Allah's peace and blessings be upon him) let them choose between the three types of Hajj. He, and so did some of his Companions, drove the sacrificial animals with him from Dhu al-Hulayfah, while others did not do so. When they approached Makkah, some of those who brought the sacrificial animals with them intending to do Ifrād or Qirān Hajj, suggested that they should revoke Hajj and replace it with ‘Umrah. But when they performed Tawāf and Sa‘i, the Prophet (may Allah's peace and blessings be upon him) ordered them emphatically to shorten their hair, end the state of Ihrām for ‘Umrah, assume Ihrām for Hajj, and offer the sacrificial animals in order for them to perform both Hajj and ‘Umrah on a single journey. He ordered those who could not afford to offer a sacrificial animal to fast ten days: three during the Hajj days, starting from the time they assume Ihrām for ‘Umrah, and seven when they would return home.
When the Prophet (may Allah's peace and blessings be upon him) arrived at Makkah, he touched the Black Stone and performed seven rounds of Tawāf. He walked energetically during the first three rounds, which represented the Tawāf of Arrival, and normally in the remaining four rounds. Then he offered two Rak‘ahs behind the Station of Ibrahīm, headed towards Mount Safa, and performed the seven-round Sa‘i between Safa and Marwah. He would run between the two pillars and walk in a normal pace elsewhere. The Prophet (may Allah's peace and blessings be upon him) did not end his Ihrām until he finished Hajj and slaughtered his sacrificial animals on the Day of Nahr. After he had finished Hajj, had thrown the ‘Aqabah pebbles, and had slaughtered his sacrificial animals and shaved his head on the Day of Nahr, in what is known as the first Tahallul, he headed towards the Ka‘bah in the forenoon and performed Tawāf around it. Then he did all the things that were prohibited under Ihrām, including having sexual relations with his wives. Those among his Companions who had driven the sacrificial animals with them followed suit.