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A194959
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Fractalization of (1+[n/2]), where [ ]=floor.
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55
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1, 1, 2, 1, 3, 2, 1, 3, 4, 2, 1, 3, 5, 4, 2, 1, 3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 1, 3, 5, 7, 8, 6, 4, 2, 1, 3, 5, 7, 9, 8, 6, 4, 2, 1, 3, 5, 7, 9, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 12, 10, 8, 6, 4, 2, 1, 3, 5, 7, 9, 11, 13, 12, 10, 8, 6, 4, 2, 1, 3, 5
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OFFSET
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1,3
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COMMENTS
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Suppose that p(1), p(2), p(3), ... is an integer sequence satisfying 1<=p(n)<=n for n>=1. Define g(1)=(1) and for n>1, form g(n) from g(n-1) by inserting n so that its position in the resulting n-tuple is p(n). The sequence f obtained by concatenating g(1), g(2), g(3), ... is clearly a fractal sequence, here introduced as the fractalization of p. The interspersion associated with f is here introduced as the interspersion fractally induced by p, denoted by I(p); thus, the k-th term in the n-th row of I(p) is the position of the k-th n in f. Regarded as a sequence, I(p) is a permutation of the positive integers; its inverse permutation is denoted by Q(p).
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Example: Let p=(1,2,2,3,3,4,4,5,5,6,6,7,7,...)=A008619. Then g(1)=(1), g(2)=(1,2), g(3)=(1,3,2), so that
f=(1,1,2,1,3,2,1,3,4,2,1,3,5,4,2,1,3,5,6,4,2,1,...)=A194959; and I(p)=A057027, Q(p)=A064578.
The interspersion I(P) has the following northwest corner, easily read from f:
1...2...4...7...11..16..22
3...6...10..15..21..28..36
5...8...12..17..23..30..38
9...14..20..27..35..44..54
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Following is a chart of selected p, f, I(p), and Q(p):
p.........f.........I(p)......Q(p)
A000027...A002260...A000027...A000027
A008619...A194959...A057027...A064578
A194960...A194961...A194962...A194963
A053824...A194965...A194966...A194967
A053737...A194973...A194974...A194975
A019446...A194968...A194969...A194970
A049474...A194976...A194977...A194978
A194979...A194980...A194981...A194982
A194964...A194983...A194984...A194985
A194986...A194987...A194988...A194989
Count odd numbers up to n, then even numbers down from n. - Franklin T. Adams-Watters, Jan 21 2012
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REFERENCES
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Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.
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LINKS
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Table of n, a(n) for n=1..94.
Wikipedia, Fractal sequence
MathWorld, Fractal sequence
MathWorld, Interspersion
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EXAMPLE
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The sequence p=A008619 begins with 1,2,2,3,3,4,4,5,5,..., so that g(1)=(1). To form g(2), write g(1) and append 2 so that in g(2) this 2 has position p(2)=2: g(2)=(1,2). Then form g(3) by inserting 3 at position p(3)=2: g(3)=(1,3,2), and so on. The fractal sequence A194959 is formed as the concatenation g(1)g(2)g(3)g(4)g(5)...=(1,1,2,1,3,2,1,3,4,2,1,3,5,4,2,...).
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MATHEMATICA
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r = 2; p[n_] := 1 + Floor[n/r]
Table[p[n], {n, 1, 90}] (* A008619 *)
g[1] = {1}; g[n_] := Insert[g[n - 1], n, p[n]]
f[1] = g[1]; f[n_] := Join[f[n - 1], g[n]]
f[20] (* A194959 *)
row[n_] := Position[f[30], n];
u = TableForm[Table[row[n], {n, 1, 5}]]
v[n_, k_] := Part[row[n], k];
w = Flatten[Table[v[k, n - k + 1], {n, 1, 13},
{k, 1, n}]] (* A057027 *)
q[n_] := Position[w, n]; Flatten[
Table[q[n], {n, 1, 80}]] (* A064578 *)
Flatten[FoldList[Insert[#1, #2, Floor[#2/2] + 1] &, {}, Range[10]]] (* Birkas Gyorgy, Jun 30 2012 *)
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CROSSREFS
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Cf. A008619, A057027, A064578; A194029 (introduces the natural fractal sequence and natural interspersion of a sequence - different from those introduced at A194959.)
Sequence in context: A211189 A194968 A194980 * A194921 A195079 A124458
Adjacent sequences: A194956 A194957 A194958 * A194960 A194961 A194962
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling, Sep 06 2011
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EXTENSIONS
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Name corrected by Franklin T. Adams-Watters, Jan 21 2012
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STATUS
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approved
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