cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A194037 Inverse permutation to A194036; contains every positive integer exactly once.

Original entry on oeis.org

1, 3, 6, 10, 15, 2, 5, 9, 14, 20, 21, 28, 4, 8, 13, 19, 26, 27, 35, 36, 45, 7, 12, 18, 25, 33, 34, 43, 44, 54, 55, 66, 11, 17, 24, 32, 41, 42, 52, 53, 64, 65, 77, 78, 91, 16, 23, 31, 40, 50, 51, 62, 63
Offset: 0

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Crossrefs

Cf. A194036.

Programs

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Aug 12 2011

Keywords

Comments

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

Examples

			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)

References

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

Links

Crossrefs

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

Programs

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

Formula

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

Extensions

Edited by M. F. Hasler, Apr 23 2022
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