Here are a few more self-descriptive fractal sequences. "Upper trimming" is another technique for determining if a sequence is a fractal sequence, if the sequence contains all positive integers. The first occurrence of each integer is deleted from the sequence. If the resulting sequence is the same is the original, then it is a fractal sequence. This is an example of a fractal sequence: 1, 1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 10, 11, 12, 4, 13, 14, 2, ... If we denote the first occurrence of each integer by X, we get: X, 1, X, 1, X, X, 2, X, 1, X, X, X, 3, X, X, X, X, 4, X, X, 2, ... and dropping the Xs: 1, 1, 2, 1, 3, 4, 2, ... Which is the beginning of the original sequence. This sequence is also self-descriptive, in that each element gives the number of Xs (first occurrences of integers) that were removed just before it. Here are the first 200 terms: 1, 1, 2, 1, 3, 4, 2, 5, 1, 6, 7, 8, 3, 9, 10, 11, 12, 4, 13, 14, 2, 15, 16, 17, 18, 19, 5, 20, 1, 21, 22, 23, 24, 25, 26, 6, 27, 28, 29, 30, 31, 32, 33, 7, 34, 35, 36, 37, 38, 39, 40, 41, 8, 42, 43, 44, 3, 45, 46, 47, 48, 49, 50, 51, 52, 53, 9, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 10, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 11, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 12, 87, 88, 89, 90, 4, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 13, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 14, 118, 119, 2, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 15, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 16, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 17, 168, 169, 170, 171, 172, 173, 174, 175, 176 Here's another one: 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 4, 6, 3, 5, 7, 2, 8, 1, 4, 6, 3, 9, 5, 7, 2, 10, ... This sequence gives the number of numbers that are retained between Xs that are dropped. Alternatively, each element is the number of numbers between two first occurrences of integers. For example, the first 3 above describes the three numbers 2, 1, 4 between the first 5 and the first 6. Here are the first 200 terms: 1, 1, 2, 1, 3, 2, 1, 4, 3, 5, 2, 1, 4, 6, 3, 5, 7, 2, 8, 1, 4, 6, 3, 9, 5, 7, 2, 10, 8, 1, 4, 6, 3, 11, 9, 5, 12, 7, 13, 2, 10, 8, 1, 14, 4, 6, 3, 11, 9, 5, 15, 12, 7, 13, 16, 2, 10, 8, 1, 14, 17, 4, 6, 3, 11, 9, 5, 15, 18, 12, 7, 19, 13, 16, 2, 10, 8, 1, 14, 17, 20, 4, 21, 6, 3, 11, 9, 22, 5, 15, 18, 12, 7, 19, 23, 13, 16, 2, 24, 10, 8, 1, 14, 17, 20, 4, 21, 6, 25, 3, 11, 9, 22, 5, 26, 15, 18, 12, 7, 19, 23, 13, 27, 16, 2, 28, 24, 10, 8, 1, 14, 17, 20, 4, 21, 6, 29, 25, 3, 11, 9, 22, 5, 26, 15, 30, 18, 31, 12, 7, 19, 23, 32, 13, 27, 16, 2, 28, 24, 33, 10, 8, 1, 34, 14, 17, 20, 4, 21, 6, 29, 25, 3, 11, 9, 35, 22, 5, 26, 15, 30, 18, 31, 12, 7, 36, 19, 23, 32, 13, 27, 37, 16, 2, 28, 24, 33, 10, 8, 1 This last fractal sequence describes the number of Xs that are dropped and the number of numbers written between dropped Xs: 1, 1, 2, 1, 3, 4, 2, 1, 5, 3, 6, 7, 8, 4, 2, 1, 9, 10, 11, 12, 5, 3, 6, 7, 13, 14, 8, 4, 15, 2, 16, 17, 18, 19, 20, 1, 9, 10, 11, 12, 21, 22, 23, 5, 3, 6, 24, 25, 26, 27, 28, 29, 7, 13, 14, 8, 4, 15, 30, 31, 32, 33, 34, 35, 36, 2, 16, 17, 18, 19, 20, 1, 37, 38, 39, 40, 41, 42, 43, 44, 9, 10, 11, 12, 21, 22, 23, 5, 45, 46, 47, 48, 3, 6, 24, 25, 49, 50, 26, 27, 51, 28, 52, 53, 54, 55, 56, 57, 58, 59, 60, 29, 7, 13, 14, 8, 4, 15, 30, 31, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 32, 33, 34, 35, 36, 2, 16, 17, 18, 19, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 20, 1, 37, 38, 39, 40, 41, 42, 43, 44, 9, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 10, 11, 12, 21, 22, 23, 5, 45, 46, 47, 48, 3, 94, 95, 96, 97, 98, 6, 24, 25, 49, 50, 99, 100, 101, 26 Kerry