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A088370
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Triangle T(n,k), read by rows, where the n-th row is a binary arrangement of the numbers 1 through n.
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5
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1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 5, 3, 2, 4, 1, 5, 3, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| The n-th row differs from the prior row only by the presence of n. See A088371 for the positions in the n-th row that n is inserted.
Comment from Clark Kimberling, Aug 02 2007: (Start)
At A131966, this sequence is cited as the fractal sequence of the Cantor set C.
Recall that C is the set of fractions in [0,1] whose base 3 representation consists solely of 0s and 2s.
Arrange these fractions as follows:
0
0, .2
0, .02, .2
0, .02, .2, .22
0, .002, .02, .2, .22, etc.
Replace each number x by its order of appearance, counting each distinct predecessor of x only once, getting
1;
1, 2;
1, 3, 2;
1, 3, 2, 4;
1, 5, 3, 2, 4;
Concatenate these to get the current sequence, which is a fractal sequence as defined in "Fractal sequences and interspersions".
One property of such a sequence is that it properly contains itself as a subsequence (infinitely many times). (End)
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REFERENCES
| Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.
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LINKS
| Alois P. Heinz, Rows n = 1..141, flattened
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FORMULA
| T(n,n) = 2^(floor(log(n)/log(2))). Construction. The 2n-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n, after multiplying each term by 2. The (2n-1)-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n-1, after multiplying each term by 2.
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EXAMPLE
| Row 5 is formed from row 3, {1,3,2} and row 2, {1,2}: {1,5,3,2,4} = {1*2-1,3*2-1,2*2-1}|{1*2,2*2}.
Rows are: {1}, {1, 2}, {1, 3, 2}, {1, 3, 2, 4}, {1, 5, 3, 2, 4}, {1, 5, 3, 2, 6, 4}, {1, 5, 3, 7, 2, 6, 4}, ...
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MAPLE
| T:= proc(n) option remember;
`if` (n=1, 1, [map(x-> 2*x-1, [T(n-iquo(n, 2))])[],
map(x-> 2*x, [T( iquo(n, 2))])[]][])
end:
seq (T(n), n=1..20); # Alois P. Heinz, Oct 28 2011
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CROSSREFS
| Cf. A088371.
Diagonal gives A053644. Cf. A049773. - Alois P. Heinz, Oct 28 2011
Sequence in context: A085014 A082074 A132283 * A113787 A115624 A076291
Adjacent sequences: A088367 A088368 A088369 * A088371 A088372 A088373
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Sep 28 2003
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