A074294 -id:A074294 - cp's OEIS frontend

A194029 Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).

1, 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, 30, 31, 32, 33, 34

Offset: 1

Clark Kimberling, Aug 12 2011

Suppose that c(1), c(2), c(3), ... is an increasing sequence of positive integers with c(1) = 1, and that the sequence c(k+1) - c(k) is strictly increasing. The natural fractal sequence f of c is defined by:
  If c(k) <= n < c(k+1), then f(n) = 1 + n - c(k).
This defines the present sequence a(n) = f(n) for c = A000045.
The natural interspersion of c is here introduced as the array given by T(n,k) =(position of k-th n in f).  Note that c = (row 1 of T).
As a different example from the one considered here (c = A000045), let c = A000217 = (1, 3, 6, 10, 15, ...), the triangular numbers, so that f = (1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, ...) = A002260, and a northwest corner of T = A194029 is:
                    1   3   6  10  15  ...
                    2   4   7  11  16  ...
                    5   8  12  17  23  ...
                    9  13  18  24  31  ...
                   ...
Since every number in the set N of positive integers occurs exactly once in this (and every) interspersion, a listing of the terms of T by antidiagonals comprises a permutation, p, of N; letting q denote the inverse of p, we thus have for each c a fractal sequence, an interspersion T, and two permutations of N:
       c         f       T / p       q
    A000045   A194029   A194030   A194031
    A000290   A071797   A194032   A194033
    A000217   A002260   A066182   A066181
    A028387   A074294   A194034   A194035
    A028872   A071797   A194036   A194037
    A034856   A002260   A194038   A194040
It appears that this is also a triangle read by rows in which row n lists the first A000045(n) positive integers, n >= 1 (see example). - Omar E. Pol, May 28 2012
This is true, because the sequence c = A000045 has the property that c(k+1)  - c(k) = c(k-1), so the number of integers {1, 2, 3, ...} to be filled in from index n = c(k) to n = c(k+1)-1 is equal to c(k-1); see also the first EXAMPLE. - M. F. Hasler, Apr 23 2022

			The sequence (1, 2, 3, 5, 8, 13, ...) is used to place '1's in positions numbered 1, 2, 3, 5, 8, 13, ...  Then gaps are filled in with consecutive counting numbers:
  1, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 5, 1, ...
From _Omar E. Pol_, May 28 2012: (Start)
Written as an irregular triangle the sequence begins:
  1;
  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; ...
The row lengths are A000045(n).
(End)

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

Alois P. Heinz, Rows n = 1..20, flattened
Clark Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.

Cf. A000045 (Fibonacci numbers).
Cf. A066628, A194030, A194031 (natural interspersion of A000045 and inverse permutation).
Cf. A130853.

T:= n-> $1..(<<0|1>, <1|1>>^n)[1, 2]:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 11 2024

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 *)
Flatten[Range[Fibonacci[Range[66]]]] (* Birkas Gyorgy, Jun 30 2012 *)

a(n) = A066628(n)+1. - Alan Michael Gómez Calderón, Oct 30 2023

Edited by M. F. Hasler, Apr 23 2022

A071797 Restart counting after each new odd integer (a fractal sequence).

Original entry on oeis.org

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

Offset: 1

Antonio Esposito, Jun 06 2002

The following sequences all have the same parity: A004737, A006590, A027052, A071028, A071797, A078358, A078446.
This is also a triangle read by rows in which row n lists the first 2*n-1 positive integers, n >= 1 (see example). - Omar E. Pol, May 29 2012
a(n) mod 2 = A071028(n). - Boris Putievskiy, Jul 24 2013
The triangle in the example is the triangle used by Kircheri in 1664. See the link "Mundus Subterraneus". - Charles Kusniec, Sep 11 2022

			a(1)=1; a(9)=5; a(10)=1;
From _Omar E. Pol_, May 29 2012: (Start)
Written as a triangle the sequence begins:
  1;
  1, 2, 3;
  1, 2, 3, 4, 5;
  1, 2, 3, 4, 5, 6, 7;
  1, 2, 3, 4, 5, 6, 7, 8, 9;
  1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11;
  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;
Row n has length 2*n - 1 = A005408(n-1). (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
C. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117.
Athanasii Kircheri, Mundus Subterraneus, (1664), pg. 24.
F. Smarandache, Only Problems, Not Solutions!, Phoenix,AZ: Xiquan,1993.
Michael Somos, Sequences used for indexing triangular or square arrays

Cf. A002260, A004737, A006590, A027052, A071028, A078358, A078446.
Cf. A074294.
Row sums give positive terms of A000384.

import Data.List (inits)
a071797 n = a071797_list !! (n-1)
a071797_list = f $ tail $ inits [1..] where
   f (xs:_:xss) = xs ++ f xss
-- Reinhard Zumkeller, Apr 14 2014

A071797 := proc(n)
    n-A048760(n-1) ;
end proc: # R. J. Mathar, May 29 2016

Array[Range[2# - 1]&, 10] // Flatten (* Jean-François Alcover, Jan 30 2018 *)

PARI
```
a(n)=if(n<1,0,n-sqrtint(n-1)^2)
```

a(n) = 1 + A053186(n-1).
a(n) = n - 1 - ceiling(sqrt(n))*(ceiling(sqrt(n))-2); n > 0.
a(n) = n - floor(sqrt(n-1))^2, distance between n and the next smaller square. - Marc LeBrun, Jan 14 2004

A179753 Integers (2k)-1..0 followed by integers (2k)+1..0 and so on.

Original entry on oeis.org

1, 0, 3, 2, 1, 0, 5, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0

Offset: 1

Antti Karttunen, Aug 03 2010

nonn

Flatten[Table[Range[2n-1,0,-1],{n,10}]] (* Harvey P. Dale, Oct 08 2012 *)

from math import isqrt
def A179753(n): return (k:=(m:=isqrt(n))+(n>m*(m+1)))*(1+k)-n # Chai Wah Wu, Jun 06 2025

a(n) = (2*A000194(n)) - A074294(n).

A035505 Active part of Kimberling's expulsion array as a triangular array.

Original entry on oeis.org

4, 2, 6, 2, 7, 4, 8, 7, 9, 2, 10, 6, 6, 2, 11, 9, 12, 7, 13, 8, 13, 12, 8, 9, 14, 11, 15, 2, 16, 6, 2, 11, 16, 14, 6, 9, 17, 8, 18, 12, 19, 13, 18, 17, 12, 9, 19, 6, 13, 14, 20, 16, 21, 11, 22, 2, 16, 14, 21, 13, 11, 6, 22, 19, 2, 9, 23, 12, 24, 17, 25, 18, 23, 2, 12, 19, 24, 22, 17, 6

Offset: 1

N. J. A. Sloane

Active or shuffle part of Kimberling's expulsion array (A035486) is given by the elements K(i,j), where j < 2*i-3. [Enrique Pérez Herrero, Apr 14 2010]

			4 2; 6 2 7 4; 8 7 9 2 10 6; ...

R. K. Guy, Unsolved Problems Number Theory, Sect. E35.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000
Clark Kimberling, Problem 1615, Crux Mathematicorum, Vol. 17 (2) 44 1991; Solution to Problem 1615, Crux Mathematicorum, Vol. 18, March 1992, p. 82-83.

Cf. A006852, A007063, A038807, A035486.

Cf. A175312, A074294, A000194, A006852, A007063. [Enrique Pérez Herrero, Apr 14 2010]

A000194[n_] := Floor[(1 + Sqrt[4 n - 3])/2];
A074294[n_] := n - 2*Binomial[Floor[1/2 + Sqrt[n]], 2];
K[i_, j_] := i + j - 1 /; (j >= 2 i - 3);
K[i_, j_] := K[i - 1, i - (j + 2)/2] /; (EvenQ[j] && (j < 2 i - 3));
K[i_, j_] := K[i - 1, i + (j - 1)/2] /; (OddQ[j] && (j < 2 i - 3));
A035505[n_] := K[A000194[n] + 2, A074294[n]]
(* Enrique Pérez Herrero, Apr 14 2010 *)

From Enrique Pérez Herrero, Apr 14 2010: (Start)
a(n) = K(A000194(n)+2, A074294(n)), where
  K(i,j) = i + j - 1; (j >= 2*i - 3)
  K(i,j) = K(i-1, i-(j+2)/2) if j is even and j < 2*i - 3
  K(i,j) = K(i-1, i+(j-1)/2); if j is odd and j < 2*i - 3.
(End)

More terms from James Sellers, Dec 23 1999

A194011 Natural interspersion of A002061; a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 3, 2, 7, 4, 5, 13, 8, 9, 6, 21, 14, 15, 10, 11, 31, 22, 23, 16, 17, 12, 43, 32, 33, 24, 25, 18, 19, 57, 44, 45, 34, 35, 26, 27, 20, 73, 58, 59, 46, 47, 36, 37, 28, 29, 91, 74, 75, 60, 61, 48, 49, 38, 39, 30, 111, 92, 93, 76, 77, 62, 63, 50, 51, 40, 41, 133, 112

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, A194011 is a permutation of the positive integers; its inverse is A194012.

			Northwest corner:
1...3...7...13...21...31
2...4...8...14...22...32
5...9...15..23...33...45
6...10..16..24...34...46
11..17..25..35...47...61

Cf. A194029, A002061, A074294, A194012.

z = 40;
c[k_] := k^2 - k + 1
c = Table[c[k], {k, 1, z}]  (* A002061 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 800}]  (* A074294 *)
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}]]  (* A194011 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 80}]]  (* A194012 *)

A194034 Natural interspersion of A028387, a rectangular array, by antidiagonals.

Original entry on oeis.org

1, 5, 2, 11, 6, 3, 19, 12, 7, 4, 29, 20, 13, 8, 9, 41, 30, 21, 14, 15, 10, 55, 42, 31, 22, 23, 16, 17, 71, 56, 43, 32, 33, 24, 25, 18, 89, 72, 57, 44, 45, 34, 35, 26, 27, 109, 90, 73, 58, 59, 46, 47, 36, 37, 28, 131, 110, 91, 74, 75, 60, 61, 48, 49, 38, 39, 155, 132

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, A194034 is a permutation of the positive integers; its inverse is A194035.

			Northwest corner:
1...5...11...19...29...41
2...6...12...20...30...42
3...7...13...21...31...43
4...8...14...22...32...44
9...15..23...33...45...59

Cf. A194029, A194035.

z = 30;
c[k_] := k^2 + k - 1;
c = Table[c[k], {k, 1, z}]  (* A028387 *)
f[n_] := If[MemberQ[c, n], 1, 1 + f[n - 1]]
f = Table[f[n], {n, 1, 255}]  (* A074294 *)
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, 13}, {k, 1, n}]]  (* A194034 *)
q[n_] := Position[p, n]; Flatten[Table[q[n], {n, 1, 70}]]  (* A194035 *)

A253515 Count down from 2k to 1, then from 2(k+1) to 1 and so on.

Original entry on oeis.org

2, 1, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Offset: 1

Mikael Aaltonen, Jan 03 2015

nonn

Cf. A074294.

a253515[n_] := Block[{f}, f[x_] := 1 - x + Floor[Sqrt[x] + 3/2] * Floor[Sqrt[x] + 1/2]; Array[f, n]]; a253515[100] (* Michael De Vlieger, Jan 03 2015 *)
Flatten[Table[Range[2n,1,-1],{n,10}]] (* Harvey P. Dale, Jan 15 2016 *)

a(n) = 1 - n + floor(sqrt(n)+3/2)*floor(sqrt(n)+1/2).

a(n) = A130829(n) - A074294(n).

A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Aug 29 2024

A208233 presents an algorithm for generating permutations, where each generated permutation is self-inverse.

The sequence in each column k possesses two properties: it is both a self-inverse permutation and an intra-block permutation of natural numbers.

			Table begins:
    k=    1   2   3   4   5   6
  -----------------------------------
  n= 1:   1,  1,  3,  1,  5,  1, ...
  n= 2:   2,  2,  2,  3,  2,  5, ...
  n= 3:   3,  3,  1,  2,  3,  3, ...
  n= 4:   6,  5,  4,  4,  4,  4, ...
  n= 5:   5,  4,  8,  5,  1,  2, ...
  n= 6:   4,  6,  6, 11,  6,  6, ...
  n= 7:   7,  7,  7,  7, 14,  7, ...
  n= 8:   9, 11,  5,  9,  8, 17, ...
  n= 9:   8,  9,  9,  8, 12,  9, ...
  n= 10: 10, 10, 18, 10, 10, 15, ...
  n= 11: 15,  8, 11,  6, 11, 11, ...
  n= 12: 12, 12, 16, 12,  9, 13, ...
  n= 13: 13, 13, 13, 13, 13, 12, ...
  n= 14: 14, 19, 14, 23,  7, 14, ...
  n= 15: 11, 15, 15, 15, 15, 10, ...
  n= 16: 16, 17, 12, 21, 30, 16, ...
  n= 17: 20, 16, 17, 17, 17,  8, ...
  n= 18: 18, 18, 10, 19, 28, 18, ...
     ... .
In column 3, the first 3 blocks have lengths 3,6 and 9. In column 6, the first 2 blocks have lengths 6 and 12. Each block is a permutation of the numbers of its constituents.
The first 6 antidiagonals are:
  1;
  2,1;
  3,2,3;
  6,3,2,1;
  5,5,1,3,5;
  4,4,4,2,2,1;

Boris Putievskiy, Table of n, a(n) for n = 1..9870
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000194, A002024, A002260, A003991, A062707, A074294, A093178, A111651, A124625, A188568, A208233, A371355.

T[n_,k_]:=Module[{L,R,P,result},L=Ceiling[(Sqrt[8*n*k+k^2]-k)/(2*k)]; R=n-k*(L-1)*L/2; P=(((-1)^Max[R,k*L+1-R]+1)*R-((-1)^Max[R,k*L+1-R]-1)*(k*L+1-R))/2; result=P+k*(L-1)*L/2]
Nmax=18; Table[T[n,k],{n,1,Nmax},{k,1,Nmax}]

T(n,k) = P(n,k) + k*(L(n,k)-1)*L(n,k)/2 = P(n,k) + A062707(L(n-1),k), where L(n,k) = ceiling((sqrt(8*n*k+k^2)-k)/(2*k)), R(n,k) = n-k*(L(n,k)-1)*L(n,k)/2, P(n,k) = (((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))+1)*R(n,k)-((-1)^max(R(n,k),k*L(n,k)+1-R(n,k))-1)*(k*L(n,k)+1-R(n,k)))/2.

T(n,1) = A188568(n). T(1,k) = A093178(k). T(n,n) = A124625(n). L(n,1) = A002024(n). L(n,2) = A000194(n). L(n,3) = A111651(n). L(n,4) = A371355(n). R(n,1) = A002260(n). R(n,2) = A074294(n).

cp's OEIS Frontend

A194029 Natural fractal sequence of the Fibonacci sequence (1, 2, 3, 5, 8, ...).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A071797 Restart counting after each new odd integer (a fractal sequence).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A179753 Integers (2k)-1..0 followed by integers (2k)+1..0 and so on.

Views

Author

Keywords

Programs

Mathematica

Python

Formula

A035505 Active part of Kimberling's expulsion array as a triangular array.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A194011 Natural interspersion of A002061; a rectangular array, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A194034 Natural interspersion of A028387, a rectangular array, by antidiagonals.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A253515 Count down from 2*k to 1, then from 2*(k+1) to 1 and so on.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A375797 Table T(n, k) read by upward antidiagonals. The sequences in each column k is a triangle read by rows (blocks), where each row is a permutation of the numbers of its constituents. Row number n in column k has length n*k = A003991(n,k); see Comments.

Views

Author

Keywords

A253515 Count down from 2k to 1, then from 2(k+1) to 1 and so on.