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A364880
a(0) = 0. If a(n-1) is a novel term, a(n) = a(a(n-1)); otherwise a(n) is the number of distinct terms occurring prior to a(n-1) which are <= a(n-1).
1
0, 0, 1, 0, 1, 2, 1, 2, 3, 0, 1, 2, 3, 4, 1, 2, 3, 4, 5, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 2, 3, 4, 5, 6, 7, 8, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 3, 4, 5, 6, 7, 8, 9
OFFSET
0,6
COMMENTS
The novel terms are all records so the records subsequence is A000027.
Following each record term r, the sequence resets to a(r) then increments to novel term r+1, which introduces a(r+1), and so on (compare with A002262, which resets to 0 after each record).
The proper subsequence {a(a(r))} is a copy of the original, which therefore is fractal.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..10353 (rows 0..150, flattened)
Michael De Vlieger, Scatterplot of a(n), n = 0..119 showing the first terms in each row in red, and all other terms in dark blue, labeling these respectively in the same color so as to demonstrate T(n,1) = a(n-1), where T(n,1) is the first term of row n, and sequence a represents the flattened irregular table T.
Michael De Vlieger, Scatterplot of a(n), n = 0..2047 with the same color approach used above, but only labeling T(n,1).
EXAMPLE
a(0) = 0 is a novel term, so a(1) = a(a(0)) = a(0) = 0.
a(2) = 1 because there is just one distinct prior term (0) which is <= 0.
Since a(2) = 1 is a novel term, a(3) = a(a(2)) = a(1) = a(0) = 0.
a(4) = 1 because a(3) = 0 is a repeat term and there is only one distinct term (0) <= 0.
The sequence can be represented as an irregular table in which each row starts with a record term, and ends with the first repeat of the same number. The first column is A000027, and the second column is the sequence itself:
The table starts:
0, 0;
1, 0, 1;
2, 1, 2;
3, 0, 1, 2, 3;
4, 1, 2, 3, 4;
5, 2, 3, 4, 5;
6, 1, 2, 3, 4, 5, 6;
7, 2, 3, 4, 5, 6, 7;
8, 3, 4, 5, 6, 7, 8;
9, 0, ...
MATHEMATICA
nn = 120; a[0] = j = k = 0; r = -1; Do[If[j > r, k = a[j]; r = j, k++]; Set[{a[n], j}, {k, k}], {n, nn}]; Array[a, nn + 1, 0] (* Michael De Vlieger, Aug 14 2023 *)
CROSSREFS
Sequence in context: A348910 A123590 A092872 * A141455 A292627 A113125
KEYWORD
nonn,easy,tabf
AUTHOR
STATUS
approved