

A132224


A dense normalized fractal sequence.


3



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, 5, 10, 9, 1, 8, 4, 7, 2, 6, 11, 3, 5, 10, 9, 1, 8, 4, 7, 2, 12, 6, 11, 3, 5, 10, 9
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OFFSET

1,3


COMMENTS

This sequence is also given by its placement sequence, A132226.
Needs bfile.


REFERENCES

C. Kimberling, Proper selfcontaining sequences, fractal sequences and parasequences, preprint, 2007.


LINKS

Table of n, a(n) for n=1..78.
Clark Kimberling, Proper selfcontaining sequences, fractal sequences and parasequences, unpublished manuscript, 2007, cached copy, with permission.


FORMULA

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 nth 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.


EXAMPLE

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;


MAPLE

A132224_row := proc(n)
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:
seq(op(A132224_row(n)), n=1..12) ; # R. J. Mathar, May 08 2016


MATHEMATICA

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 *)


CROSSREFS

Cf. A132223.
Sequence in context: A264744 A143604 A021475 * A194961 A195110 A167198
Adjacent sequences: A132221 A132222 A132223 * A132225 A132226 A132227


KEYWORD

nonn


AUTHOR

Clark Kimberling, Aug 14 2007


STATUS

approved



