

A194959


Fractalization of (1+floor(n/2)).


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
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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(n1) by inserting n so that its position in the resulting ntuple 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 kth term in the nth row of I(p) is the position of the kth 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. AdamsWatters, Jan 21 2012


REFERENCES

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157168.


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. AdamsWatters, Jan 21 2012


STATUS

approved



