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EXAMPLE
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For n=1, the initial set from which we start is {1}, and it already contains 1, so a(1) = 1 as the size of that set is 1.
For n=2, the initial set is {2}, which will become set {1, 7} (because 2/2 = 1 and 3*2+1 = 7), and that set already contains 1, thus a(2) = 2.
For n=3, the initial set is {3}, the next set is 3*3+1 = {10}, from which we get {5, 31} -> {16, 94} -> {8, 47, 49, 283} -> {4, 25, 142, 148, 850} -> {2, 13, 71, 74, 76, 425, 427, 445, 2551} and from that one we get these 12 numbers: {1, 7, 37, 38, 40, 214, 223, 229, 1276, 1282, 1336, 7654}, and because 1 is among these, a(3) = 12.
For n = 12, the iteration proceeds as follows: {12} -> {6, 37} -> {3, 19, 112} -> {10, 56, 58, 337} -> {5, 28, 29, 31, 169, 175, 1012} -> {14, 16, 85, 88, 94, 506, 508, 526, 3037} -> {7, 8, 43, 44, 47, 49, 253, 254, 256, 263, 265, 283, 1519, 1525, 1579, 9112 -> {4, 22, 25, 127, 128, 130, 133, 142, 148, 760, 763, 769, 790, 796, 850, 4556, 4558, 4576, 4738, 27337} -> {2, 11, 13, 64, 65, 67, 71, 74, 76, 380, 382, 385, 391, 395, 398, 400, 425, 427, 445, 2278, 2279, 2281, 2288, 2290, 2308, 2369, 2371, 2389, 2551, 13669, 13675, 13729, 14215, 82012} -> {1, 7, 32, 34, 37, 38, 40, 190, 191, 193, 196, 199, 200, 202, 214, 223, 229, 1139, 1141, 1144, 1145, 1147, 1154, 1156, 1174, 1186, 1195, 1201, 1276, 1282, 1336, 6835, 6838, 6844, 6865, 6871, 6925, 7108, 7114, 7168, 7654, 41006, 41008, 41026, 41188, 42646, 246037}. As this last set contains 1 and has 47 members, a(12) = 47. Note how here in the 7th iteration the term 22 is a child of both 7 (as 3*7+1) and 44 (as 44/2), but as these are sets, not multisets, 22 occurs only once in {4, 22, 25, ...}.
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