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A194959 Fractalization of (1+[n/2]), where [ ]=floor. 55
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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).

...

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

...

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

REFERENCES

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

LINKS

Table of n, a(n) for n=1..94.

Wikipedia, Fractal sequence

MathWorld, Fractal sequence

MathWorld, Interspersion

EXAMPLE

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,...).

MATHEMATICA

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 *)

CROSSREFS

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

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Sep 06 2011

EXTENSIONS

Name corrected by Franklin T. Adams-Watters, Jan 21 2012

STATUS

approved

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Last modified December 4 17:35 EST 2016. Contains 278755 sequences.