A194029 -id:A194029 - cp's OEIS frontend

A194959 Fractalization of (1 + floor(n/2)).

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

Offset: 1

Clark Kimberling, Sep 06 2011

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

			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)

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

Paul Lévy, Sur quelques classes de permutations, Compositio Mathematica, Volume 8, 1951, pages 1-48. P_n = g(n).
Eric Weisstein's World of Mathematics, Fractal sequence
Eric Weisstein's World of Mathematics, Interspersion
Wikipedia, Fractal sequence

Cf. A000142, A000217, A005408, A005843, A008619, A057027, A064578, A209229, 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).

Cf. A003558 (g permutation order), A102417 (index), A330081 (on bits), A057058 (inverse).

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 *)

T(n,k) = min(k<<1-1,(n-k+1)<<1); \\ Kevin Ryde, Oct 09 2020

From Werner Schulte, May 27 2018 and Jul 10 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.
The P(n,x) satisfy the recurrence equation P(n+1,x) = P(n,x) + x^n*P(1,x) for n > 0 and initial value P(1,x) = (1+x)/(1-x).
Let B(n,k) be multiplicative with B(n,p^e) = A(n,e+1) for e >= 0 and some fixed n > 0. That yields the Dirichlet g.f.: Sum_{k>0} B(n,k)/k^s = (zeta(s))^3/(zeta(2*s)*zeta(n*s)).
Sum_{k=1..n} A(k,n+1-k)*A209229(k) = 2*n-1. (conjectured)
(End)
From Kevin Ryde, Oct 09 2020: (Start)
T(n,k) = 2*k-1 if 2*k-1 <= n, or 2*(n+1-k) if 2*k-1 > n. [Lévy, chapter 1 section 1 equations (a),(b)]
Fixed points T(n,k)=k for k=1 and k = (2/3)*(n+1) when an integer. [Lévy, chapter 1 section 2 equation (3)]
(End)

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

A122197 Fractal sequence: count up to successive integers twice.

Original entry on oeis.org

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

Offset: 1

Franklin T. Adams-Watters, Aug 25 2006

Fractal - deleting the first occurrence of each integer leaves the original sequence. Also, deleting all the 1's leaves the original sequence plus 1. New values occur at square indices. 1's occur at indices m^2+1 and m^2+m+1. Ordinal transform of A122196.
Except for its initial 1, A122197 is the natural fractal sequence of A002620; that is, A122197(n+1) is the number of the row of A194061 that contains n. See A194029 for definition of natural fractal sequence. - Clark Kimberling, Aug 12 2011
From Johannes W. Meijer, Sep 09 2013: (Start)
Triangle read by rows formed from antidiagonals of triangle A002260.
The row sums equal A008805(n-1) and the antidiagonal sums equal A211534(n+5). (End)

			The first few rows of the sequence a(n) as a triangle T(n, k):
  n/k  1   2   3
  1    1
  2    1
  3    1,  2
  4    1,  2
  5    1,  2,  3
  6    1,  2,  3

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A122196, A000290, A033638, A002260, A001477, A000194, A339399.

import Data.List (transpose, genericIndex)
a122197 n k = genericIndex (a122197_row n) (k - 1)
a122197_row n = genericIndex a122197_tabf (n - 1)
a122197_tabf = concat $ transpose [a002260_tabl, a002260_tabl]
a122197_list = concat a122197_tabf
-- Reinhard Zumkeller, Aug 07 2015, Jul 19 2012

From Johannes W. Meijer, Sep 09 2013: (Start)
a := proc(n) local t: t := floor((sqrt(4*n-3)-1)/2): (n-1) mod (t+1) + 1 end: seq(a(n), n=1..105); # End first program
T := proc(n, k): if n < 1 then return(0) elif k < 1 or k> floor((n+1)/2) then return(0) else k fi: end: seq(seq(T(n, k), k=1..floor((n+1)/2)), n=1..19); # End second program. (End)

With[{c=Table[Range[n],{n,10}]},Flatten[Riffle[c,c]]] (* Harvey P. Dale, Apr 19 2013 *)

a(n)=n - (sqrtint(4*n) + 1)\2*sqrtint(n-1) \\ Charles R Greathouse IV, Jun 08 2020

from math import isqrt
def A122197(n): return 1 if n<=1 else 1+((n-1)%((m:=isqrt(n-1))+int(n-1>m*(m+1)))) # Chai Wah Wu, Jun 05 2025

From Boris Putievskiy, Sep 09 2013: (Start)
a(n) = (A001477(n-1) mod A000194(n-1)) + 1 for n >= 2 with a(1) = 1.
a(n) = ((n-1) mod (t+1)) + 1, where t = floor((sqrt(4*n-3)-1)/2). (End)
From Johannes W. Meijer, Sep 09 2013: (Start)
T(n, k) = k for n >= 1 and 1 <= k <= (n+1)/2; T(n, k) = 0 elsewhere.
T(n, k) = A002260(n-k, k). (End)
a(n) = n - floor(sqrt(n) + 1/2)*floor(sqrt(n-1)). - Ridouane Oudra, Jun 08 2020
a(n) = A339399(2n-1). - Wesley Ivan Hurt, Jan 09 2022

A074294 Integers 1 to 2k followed by integers 1 to 2k + 2 and so on.

Original entry on oeis.org

1, 2, 1, 2, 3, 4, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 2

Offset: 1

Michael Somos, Aug 20 2002

From Cino Hilliard, Sep 13 2004: (Start)
Also the numerator of the fraction in the continued fraction expansion of sqrt(n) for nonsquare n = 2,3,5,6,7... . E.g., for n = 7,
sqrt(7).=.2.+.3................
...............4..+.3..........
.....................4..+.3....
...........................4.....
3 is the 5th entry in the table. sqrt(1) and sqrt(4) are not included because 1 and 4 are squares." (End)
A074294 is the natural fractal sequence of A002061; the corresponding natural interspersion is A194011; see A194029 for definitions. - Clark Kimberling, Aug 17 2011
It appears that this is also a triangle read by rows in which row n lists the first 2*n positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012

			From _Omar E. Pol_, May 29 2012: (Start)
Written as a triangle the sequence begins:
1, 2;
1, 2, 3, 4;
1, 2, 3, 4, 5, 6;
1, 2, 3, 4, 5, 6, 7, 8;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16;
Row n has length 2*n = A005843(n). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A002061, A194011.
Cf. A071797.
Cf. A000194, A053187.

import Data.List (inits)
a074294 n = a074294_list !! (n-1)
a074294_list = f $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

seq(seq((j-n^2-n),j=n^2+n+1..(n+1)^2+n+1),n=0..20); # Robert Israel, Jan 05 2015

A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2] (* Enrique Pérez Herrero, Apr 14 2010 *)
Table[Range[2n],{n,10}]//Flatten (* Harvey P. Dale, Oct 20 2018 *)

{a(n) = n - 2 * binomial( floor( 1/2 + sqrt(n)), 2)}

c(n) = for(x=2,n,if(issquare(x)==0,a=floor(sqrt(x));print1(x-a^2", "))) /* Cino Hilliard, Sep 13 2004 */

from math import isqrt
def A074294(n): return n+(k:=(m:=isqrt(n))+(n>m*(m+1)))*(1-k) # Chai Wah Wu, Jun 06 2025

a(n) = n - 2*binomial(floor(1/2 + sqrt(n)), 2).
a(n^2 + n) = 2*n.
a(n) = n - 2 - floor(sqrt(n)+3/2)*floor(sqrt(n)-3/2). - Mikael Aaltonen, Jan 02 2015
G.f.: x/(1-x)^2 - (2*x/(1-x))*Sum_{k>=1} k*x^(k^2+k). That sum is related to Jacobi theta functions. - Robert Israel, Jan 05 2015
a(n) = n + A000194(n) - A053187(n). - Robert Israel, Jan 05 2015

A194030 Natural interspersion of the Fibonacci sequence (1,2,3,5,8,...), a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 5, 9, 10, 11, 8, 14, 15, 16, 12, 13, 22, 23, 24, 17, 18, 21, 35, 36, 37, 25, 26, 19, 34, 56, 57, 58, 38, 39, 27, 20, 55, 90, 91, 92, 59, 60, 40, 28, 29, 89, 145, 146, 147, 93, 94, 61, 41, 42, 30, 144, 234, 235, 236, 148, 149, 95, 62, 63, 43, 31

Offset: 1

Clark Kimberling, Aug 12 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, this sequence is a permutation of the positive integers; its inverse is A194031.

			Northwest corner:
  1...2...3...5...8...13
  4...6...9...14..22..35
  7...10..15..23..36..57
  11..16..24..37..58..92
  12..17..25..38..59..93

Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened).
Index entries for sequences that are permutations of the natural numbers

Cf. A130233, A194029, A194031 (inverse).
Column 1 appears to be A345347.
Although A383977 initially has many terms in comment with this, the author of A383977 has observed that the two sequences are unrelated. - N. J. A. Sloane, May 17 2025

z = 40;
c[k_] := Fibonacci[k + 1];
c = Table[c[k], {k, 1, z}]  (* A000045 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194029 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 13}, {k, 1, n}]]  (* A194030 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194031 *)
(* Second program: *)
f[x_] := Block[{i = 1}, While[x > Fibonacci[i], i++]; i - 1]; Table[Fibonacci[f[k] + n - k + 1] + k - 1, {n, 2, 12}, {k, n - 1}] // Flatten (* Michael De Vlieger, May 17 2025 *)

T(n,k) = Fibonacci(A130233(n) + k - 1) + n - 1. - Michael De Vlieger, May 17 2025

A194100 Natural interspersion of A194126; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 6, 2, 13, 7, 3, 23, 14, 8, 4, 36, 24, 15, 9, 5, 51, 37, 25, 16, 10, 11, 69, 52, 38, 26, 17, 18, 12, 89, 70, 53, 39, 27, 28, 19, 20, 112, 90, 71, 54, 40, 41, 29, 30, 21, 138, 113, 91, 72, 55, 56, 42, 43, 31, 22, 166, 139, 114, 92, 73, 74, 57, 58, 44, 32, 33, 197

Offset: 1

Clark Kimberling, Aug 15 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194100 is a permutation of the positive integers; its inverse is A194101.

			Northwest corner:
1...6...13...23...36
2...7...14...24...37
3...8...15...25...38
4...9...16...26...39
5...10..17...27...40
11..18..28...41...56

Cf. A194029, A194126, A193042, A194101.

z = 40; g = GoldenRatio;
c[k_] := -1 + Sum[Floor[j + j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* 194126 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A193042 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194100 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194101 *)

A194102 a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.

Original entry on oeis.org

1, 3, 7, 12, 19, 27, 36, 47, 59, 73, 88, 104, 122, 141, 162, 184, 208, 233, 259, 287, 316, 347, 379, 412, 447, 483, 521, 560, 601, 643, 686, 731, 777, 825, 874, 924, 976, 1029, 1084, 1140, 1197, 1256, 1316, 1378, 1441, 1506, 1572, 1639, 1708, 1778

Offset: 1

Clark Kimberling, Aug 15 2011

nonn

The natural fractal sequence of A194102 is A194103; the natural interspersion is A194104. See A194029 for definitions.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A001951, A194029, A194103, A194104.

[(&+[Floor(k*Sqrt(2)): k in [1..n]]): n in [1..100]]; // G. C. Greubel, Jun 05 2018

a[n_]:=Sum[Floor[j*Sqrt[2]], {j, 1, n}]; Table[a[n], {n, 1, 90}]

apply( A194102(n)=sum(k=1,n,sqrtint(k^2*2)), [1..99]) \\ M. F. Hasler, Jan 16 2021

apply( {A194102(n)=if(n>1, (1+n=sqrtint(n^2*2))*n\2-A194102(n-=sqrtint(n^2\2)+1)-(1+n)*n, n)}, [1..99]) \\ M. F. Hasler, Apr 23 2022

a(n) = B*(B+1)/2 - C*(C+1) - a(C) where B = floor(sqrt(2)*n) and C = floor(B/(sqrt(2)+2)). - M. F. Hasler, Apr 23 2022

A194108 Natural interspersion of A194106; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 9, 5, 3, 15, 10, 6, 7, 23, 16, 11, 12, 8, 33, 24, 17, 18, 13, 14, 45, 34, 25, 26, 19, 20, 21, 58, 46, 35, 36, 27, 28, 29, 22, 73, 59, 47, 48, 37, 38, 39, 30, 31, 90, 74, 60, 61, 49, 50, 51, 40, 41, 32, 109, 91, 75, 76, 62, 63, 64, 52, 53, 42, 43, 129, 110

Offset: 1

Clark Kimberling, Aug 15 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194108 is a permutation of the positive integers; its inverse is A194109.

			Northwest corner:
1...4...9...15...23
2...5...10..16...24
3...6...11..17...25
7...12..18..26...36
8...13..19..27...37

Cf. A194029, A194106, A194107, A194109.

z = 40; g = Sqrt[3];
c[k_] := Sum[Floor[j*g], {j, 1, k}];
c = Table[c[k], {k, 1, z}]  (* A194106 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A194107 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 8}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 16}, {k, 1, n}]]  (* A194108 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194109 *)

A194055 Natural fractal sequence of A000071 (Fibonacci numbers minus 1).

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29

Offset: 1

Clark Kimberling, Aug 13 2011

nonn

See A194029 for definitions of natural fractal sequence and natural interspersion.

Cf. A194029, A000071, A194056, A194057.

z = 50;
c[k_] := -1 + Fibonacci[k + 2]
c = Table[c[k], {k, 1, z}] (* A000071, F(n+2)-1 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194055 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 11}, {k, 1, n}]] (* A194056 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194057 *)

A194071 Natural interspersion of A194069; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 11, 8, 9, 6, 17, 12, 13, 10, 15, 25, 18, 19, 14, 21, 16, 33, 26, 27, 20, 29, 22, 23, 43, 34, 35, 28, 37, 30, 31, 24, 55, 44, 45, 36, 47, 38, 39, 32, 41, 67, 56, 57, 46, 59, 48, 49, 40, 51, 42, 81, 68, 69, 58, 71, 60, 61, 50, 63, 52, 53, 97, 82, 83

Offset: 1

Clark Kimberling, Aug 14 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194071 is a permutation of the positive integers; its inverse is A194072.

			Northwest corner:
1...3...7...11...17
2...4...8...12...18
5...9...13..19...27
6...10..14..20...28
15..21..29..37...47

Cf. A194029, A194069, A194070, A194072.

z = 70;
c[k_] := 1 + Floor[(2/3) k^2];
c = Table[c[k], {k, 1, z}]  (* A194069 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194070 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194071 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 90}]] (* A194072 *)

A194075 Natural interspersion of A194073; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 4, 2, 7, 5, 3, 13, 8, 6, 10, 19, 14, 9, 16, 11, 28, 20, 15, 22, 17, 12, 37, 29, 21, 31, 23, 18, 25, 49, 38, 30, 40, 32, 24, 34, 26, 61, 50, 39, 52, 41, 33, 43, 35, 27, 76, 62, 51, 64, 53, 42, 55, 44, 36, 46, 91, 77, 63, 79, 65, 54, 67, 56, 45, 58, 47, 109, 92, 78

Offset: 1

Clark Kimberling, Aug 14 2011

See A194029 for definitions of natural fractal sequence and natural interspersion. Every positive integer occurs exactly once (and every pair of rows intersperse), so that as a sequence, A194075 is a permutation of the positive integers; its inverse is A194076.

			Northwest corner:
1...4...7...13...19
2...5...8...14...20
3...6...9...15...21
10..16..22..31...40
11..17..23..32...41

Cf. A194029, A194074, A194073, A194076.

z = 70;
c[k_] := 1 + Floor[(3/4) k^2];
c = Table[c[k], {k, 1, z}]  (* A194073 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 300}]   (* A194074 *)
r[n_] := Flatten[Position[f, n]]
t[n_, k_] := r[n][[k]]
TableForm[Table[t[n, k], {n, 1, 7}, {k, 1, 7}]]
p = Flatten[
  Table[t[k, n - k + 1], {n, 1, 14}, {k, 1, n}]] (* A194075 *)
q[n_] := Position[p, n]; Flatten[
 Table[q[n], {n, 1, 90}]]  (* A194076 *)

cp's OEIS Frontend

A194959 Fractalization of (1 + floor(n/2)).

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A122197 Fractal sequence: count up to successive integers twice.

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A074294 Integers 1 to 2*k followed by integers 1 to 2*k + 2 and so on.

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A194030 Natural interspersion of the Fibonacci sequence (1,2,3,5,8,...), a rectangular array, by antidiagonals.

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A194100 Natural interspersion of A194126; a rectangular array, by antidiagonals.

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A194102 a(n) = Sum_{j=1..n} floor(j*sqrt(2)); n-th partial sum of Beatty sequence for sqrt(2), A001951.

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A194108 Natural interspersion of A194106; a rectangular array, by antidiagonals.

A074294 Integers 1 to 2k followed by integers 1 to 2k + 2 and so on.