As @MarcoB suggested, looking at OEIS really did help. Then using NestList I obtained the desired result.
list1 = FoldPairList[TakeDrop, Range[21], Range[6]]
{{1}, {2, 3}, {4, 5, 6}, {7, 8, 9, 10}, {11, 12, 13, 14, 15}, {16, 17, 18, 19, 20, 21}}
list2 = NestList[Delete[#, 1] + 1 &, Table[(i - 1) (i - 2)/2 + 1, {i, 2, 7}], 5]
{{1, 2, 4, 7, 11, 16}, {3, 5, 8, 12, 17}, {6, 9, 13, 18}, {10, 14, 19}, {15, 20}, {21}}
list3 = NestList[Delete[#, 1] - 1 &, Table[i (i + 1)/2, {i, 1, 6}], 5]
{{1, 3, 6, 10, 15, 21}, {2, 5, 9, 14, 20}, {4, 8, 13, 19}, {7, 12, 18}, {11, 17}, {16}}
