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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 (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

This sequence defines the square array A(n,k), n > 0 and k > 0, read by antidiagonals and the triangle T(n,k) = A(n+1-k,k) for 1 <= k <= n read by rows (see Formula and Example). - Werner Schulte, May 27 2018

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

FORMULA

From Werner Schulte, May 27 2018: (Start)

Seen as a triangle: It seems that the triangle T(n,k) for 1 <= k <= n (see Example) is the mirror image of A210535.

Seen as a square array A(n,k) and as a triangle T(n,k):

A(n,k) = 2*k-1 for 1 <= k <= n, and A(n,k) = 2*n for 1 <= n < k.

A(n+1,k+1) = A(n,k+1) + A(n,k) - A(n-1,k) for k > 0 and n > 1.

A(n,k) = A(k,n) - 1 for n > k >= 1.

P(n,x) = Sum_{k>0} A(n,k)*x^(k-1) = (1-x^n)*(1-x^2)/(1-x)^3 for n >= 1.

Q(y,k) = Sum_{n>0} A(n,k)*y^(n-1) = 1/(1-y) for k = 1 and Q(y,k) = Q(y,1) + P(k-1,y) for k > 1.

G.f.: Sum_{n>0, k>0} A(n,k)*x^(k-1)*y^(n-1) = (1+x)/((1-x)*(1-y)*(1-x*y)).

Sum_{k=1..n} A(n+1-k,k) = Sum_{k=1..n} T(n,k) = A000217(n) for n > 0.

Sum_{k=1..n} (-1)^(k-1) * A(n+1-k,k) = Sum_{k=1..n} (-1)^(k-1) * T(n,k) = A219977(n-1) for n > 0.

Product_{k=1..n} A(n+1-k,k) = Product_{k=1..n} T(n,k) = A000142(n) for n>0.

A(n+m,n) = A005408(n-1) for n > 0 and some fixed m >= 0.

A(n,n+m) = A005843(n) for n > 0 and some fixed m > 0.

Let A_m be the upper left part of the square array A(n,k) with m rows and m columns. Then det(A_m) = 1 for some fixed m > 0.

(End)

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,...).

From Werner Schulte, May 27 2018: (Start)

This sequence seen as a square array read by antidiagonals:

  n\k: 1  2  3  4  5   6   7   8   9  10  11  12 ...

  ===================================================

   1   1  2  2  2  2   2   2   2   2   2   2   2 ... (see A040000)

   2   1  3  4  4  4   4   4   4   4   4   4   4 ... (see A113311)

   3   1  3  5  6  6   6   6   6   6   6   6   6 ...

   4   1  3  5  7  8   8   8   8   8   8   8   8 ...

   5   1  3  5  7  9  10  10  10  10  10  10  10 ...

   6   1  3  5  7  9  11  12  12  12  12  12  12 ...

   7   1  3  5  7  9  11  13  14  14  14  14  14 ...

   8   1  3  5  7  9  11  13  15  16  16  16  16 ...

   9   1  3  5  7  9  11  13  15  17  18  18  18 ...

  10   1  3  5  7  9  11  13  15  17  19  20  20 ...

  etc.

This sequence seen as a triangle read by rows:

  n\k:  1  2  3  4  5   6   7   8   9  10  11  12  ...

  ======================================================

   1    1

   2    1  2

   3    1  3  2

   4    1  3  4  2

   5    1  3  5  4  2

   6    1  3  5  6  4   2

   7    1  3  5  7  6   4   2

   8    1  3  5  7  8   6   4   2

   9    1  3  5  7  9   8   6   4   2

  10    1  3  5  7  9  10   8   6   4   2

  11    1  3  5  7  9  11  10   8   6   4   2

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

  etc.

(End)

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. A000142, A000217, A005408, A005843, A008619, A057027, A064578, A210535, A219977; A000012 ( col 1 ), A157532 ( col 2 ), A040000 ( row 1 ), A113311 ( row 2 ); 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 July 19 07:15 EDT 2018. Contains 312774 sequences. (Running on oeis4.)