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A364749
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a(1) = 1. Thereafter, if a(n-1) is a novel term a(n) = a(a(n-1)), otherwise a(n) is the number of times a(n-1) has been repeated.
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2
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1, 1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 3, 1, 7, 1, 8, 4, 1, 9, 2, 2, 3, 2, 4, 2, 5, 1, 10, 1, 11, 5, 2, 6, 1, 12, 1, 13, 6, 2, 7, 1, 14, 3, 3, 4, 3, 5, 3, 6, 3, 7, 2, 8, 1, 15, 1, 16, 7, 3, 8, 2, 9, 1, 17, 1, 18, 8, 3, 9, 2, 10, 1, 19, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 3, 10
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OFFSET
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1,4
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COMMENTS
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Based on A346175, except that this sequence has offset 1, and begins a(1) = 1. When a(n-1) is a repeated term, seen k times up to and including itself, a(n) = k-1, the number of repeats of a(n-1). A record term a(m) = r beyond a(2) arises consequent to a(m-1) = 1, and is the number of times 1 has been repeated so far. The subsequence {a(r)} recovers the original sequence, which is fractal. The records subsequence is A000027.
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LINKS
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EXAMPLE
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a(1) = 1 is a novel term, seen for the first time, so a(2) = a(a(1)) = a(1) = 1. 1 has now been repeated once so a(3) = 1. Now 1 has been repeated twice, so a(4) = 2, a novel term, meaning that a(5) = a(a(4)) = a(2) = 1.
The sequence can be represented as an irregular table wherein row n starts with the n-th record term and ends with a 1 prior to the next record term, which starts the next row. The first column of the table is the records subsequence, A000027, and the second column is a copy the sequence itself.
1, 1, 1;
2, 1;
3, 1;
4, 2, 1;
5, 1;
6, 3, 1;
7, 1;
8, 4, 1;
9, 2, 2, 3, 2, 4, 2, 5, 1;
10, 1;
11, 5, 2, 6, 1;
12, 1;
13, 6, 2, 7, 1;
14, 3, 3, 4, 3, 5, 3, 6,
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MATHEMATICA
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a[1]=1; a[n_]:=a[n]=If[(s=Count[Array[a, n-1], a[n-1]])==1, a[a[n-1]], s-1];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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