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A132224
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A dense normalized fractal sequence.
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3
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1, 1, 2, 1, 2, 3, 1, 4, 2, 3, 1, 4, 2, 3, 5, 1, 4, 2, 6, 3, 5, 1, 4, 7, 2, 6, 3, 5, 1, 8, 4, 7, 2, 6, 3, 5, 1, 8, 4, 7, 2, 6, 3, 5, 9, 1, 8, 4, 7, 2, 6, 3, 10, 5, 9, 1, 8, 4, 7, 2, 6, 11, 3, 10, 5, 9, 1, 8, 4, 7, 2, 12, 6, 11, 3, 10, 5, 9, 1, 8, 4, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9, 1, 8, 4, 14, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9, 1, 8, 15, 4, 14, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9, 1, 16, 8, 15, 4, 14, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9
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OFFSET
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1,3
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COMMENTS
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This sequence is also given by its placement sequence, A132226.
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REFERENCES
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C. Kimberling, Proper self-containing sequences, fractal sequences and para-sequences, preprint, 2007.
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LINKS
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FORMULA
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This is the normalization of A132223, formed by concatenating segments obtained from segments of A132223, as follows: segment 1 is 1. Segment 2 is 12. Segment 3 is 123, formed from the segment 1423 in A132223 by retaining only numbers up to 3. Segment 4 is 1423, formed from segment 1423 in A132223 by retaining only numbers up to 4. In general, the n-th segment of A132224 (this sequence) is a permutation of (1,2,3,...,n) formed from the first segment of A132223 that contains n by deleting from it all numbers greater than n.
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EXAMPLE
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The first 7 successive segments are 1, 12, 123, 1423, 14235, 142635.
1;
1, 2;
1, 2, 3;
1, 4, 2, 3;
1, 4, 2, 3, 5;
1, 4, 2, 6, 3, 5;
1, 4, 7, 2, 6, 3, 5;
1, 8, 4, 7, 2, 6, 3, 5;
1, 8, 4, 7, 2, 6, 3, 5, 9;
1, 8, 4, 7, 2, 6, 3, 10, 5, 9;
1, 8, 4, 7, 2, 6, 11, 3, 10, 5, 9;
1, 8, 4, 7, 2, 12, 6, 11, 3, 10, 5, 9;
1, 8, 4, 7, 13, 2, 12, 6, 11, 3, 10, 5, 9;
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MAPLE
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local r, a23r, a ;
for r from 1 do
a23r := [seq(A132223(r, k), k=1..2^r)] ;
if member(n, a23r) then
break;
end if;
end do:
a := [] ;
for i from 1 to nops(a23r) do
if op(i, a23r) <= n then
a := [op(a), op(i, a23r)] ;
end if;
end do:
a ;
end proc:
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MATHEMATICA
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dense[n_] := FoldList[Riffle[#1, Range[2^#2, 2^(#2 - 1) + 1, -1]] &, {1, 2}, Range[2, n + 1]]; densefractal[n_] := Flatten[Table[Select[dense[n][[Position[dense[n], i, 2, 1][[1, 1]]]], # <= i &], {i, 2^n}]]; densefractal[5] (* Birkas Gyorgy, Apr 20 2011 *)
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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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