A054082 -id:A054082 - cp's OEIS frontend

A064579 Inverse permutation to A054082.

2, 1, 4, 3, 6, 8, 5, 10, 7, 12, 14, 9, 16, 18, 11, 20, 13, 22, 24, 15, 26, 17, 28, 30, 19, 32, 34, 21, 36, 23, 38, 40, 25, 42, 44, 27, 46, 29, 48, 50, 31, 52, 33, 54, 56, 35, 58, 60, 37, 62, 39, 64, 66, 41, 68, 43, 70, 72, 45, 74, 76, 47, 78, 49, 80, 82, 51, 84, 86, 53, 88, 55

Offset: 1

N. J. A. Sloane, Oct 16 2001

Index entries for sequences that are permutations of the natural numbers

Cf. A054082.

A054082 := proc(nmax) local a,k,n,p ; a := [2,1] ; while nops(a) < nmax do n := nops(a)+1 : k := floor((n+1)/2) ; p := 1; while p in a do p := p+1 ; od ; if n mod 2 = 1 then a := [op(a), p+k-1] ; else a := [op(a), p] ; fi ; od ; RETURN(a) ; end: A064579 := proc(a054082) local a,n,ainv ; n := 1; a := [] ; while member(n,a054082,'ainv') do a := [op(a),ainv] ; n := n+1; od; RETURN(a) ; end: a054082 := A054082(200) : a064579 := A064579(a054082) : print(op(a064579)) ; # R. J. Mathar, Jun 27 2007

a[n_] := If[OddQ[n], Floor[((n+1)/2 - 1) GoldenRatio] + (n+1)/2 + 1, Floor[(n/2 - 1) GoldenRatio] + 2]; a[2] = 1;
Sort[Array[{a[#], #}&, 100]][[All, 2]] (* Jean-François Alcover, Apr 01 2020 *)

Corrected and extended by R. J. Mathar, Jun 27 2007

A054083 a(n) = order of n in the permutation A054082 of the natural numbers if this order exists; a(n) = -1 otherwise.

Original entry on oeis.org

2, 2, 2, 2, 7, 7, 7, 7, 7, 7, 9, 7, 9, 9, 9, 9, 9, 9, -1, 9, -1, 9, -1, -1, -1, -1, 16, -1, 16, -1, 16, -1, -1, 16, 16, 16, 16, 16, -1, -1, 16, 16, -1, 16, -1, 16, -1, -1, 16, -1, -1, 16, -1, 16, -1, -1, -1, -1, -1, -1, -1, -1, -1, 16, -1, -1, -1, -1, -1, -1

Offset: 1

Clark Kimberling

sign

From Peter J. C. Moses, Jan 26 2022: (Start)
For n up to 60000,
a(n) = 2 for n = 1, 2, 3, 4;
a(n) = 7 for n = 5, 6, 7, 8, 9, 10, 12
a(n) = 9 for n = 11, 13, 14, 15, 16, 17, 18, 20, 22
a(n) = 16 for n = 27, 29, 31, 34, 35, 36, 37, 38, 41, 42, 44, 46, 49, 52, 54, 64
a(n) = 25 for n = 267, 283, 330, 343, 350, 371, 385, 393, 408, 424, 449, 467, 476, 486, 495, 504, 515, 524, 545, 578, 588, 612, 648, 674, 714. (End)

			5 -> 7 -> 9 -> 12 -> 10 -> 8-> 6-> 5, so that 5 has order 7.

Cf. A054082, A054085.

a054082[n_] := a054082[n] = If[OddQ[n], Floor[((n + 1)/2 - 1) GoldenRatio] + (n + 1)/2 + 1,
Floor[(n/2 - 1) GoldenRatio] + 2]; a054082[2] = 1;
Array[a054082[#] &, 40]  (* after Jean-François Alcover *)
Table[Length[NestWhileList[a054082, a054082[n], # != n &, 1,
    10000]] /. (10001 -> -1), {n, 1, 500}]
(* Peter J. C. Moses, Jan 26 2022 *)

Data truncated by Sean A. Irvine, Jan 23 2022

Edited by Clark Kimberling, Jan 26 2022

A054084 Permutation of N: for each k >= 1, let p(k)=least natural number not already an a(i), q(k)=p(k)+k, a(2k-1)=q(k), a(2k)=p(k).

Original entry on oeis.org

2, 1, 5, 3, 7, 4, 10, 6, 13, 8, 15, 9, 18, 11, 20, 12, 23, 14, 26, 16, 28, 17, 31, 19, 34, 21, 36, 22, 39, 24, 41, 25, 44, 27, 47, 29, 49, 30, 52, 32, 54, 33, 57, 35, 60, 37, 62, 38, 65, 40, 68, 42, 70, 43, 73, 45, 75, 46, 78, 48, 81, 50, 83

Offset: 1

Clark Kimberling

Index entries for sequences that are permutations of the natural numbers

Cf. A054082, A054085.

Odd-indexed terms: A001950 (Upper Wythoff sequence). Even-indexed terms: A000201 (Lower Wythoff sequence). Inverse permutation: A064786.

from math import isqrt
def A054084(n): return ((m:=n+1>>1)+isqrt(5*m**2)>>1)+m*(n&1) # Chai Wah Wu, Aug 25 2022

cp's OEIS Frontend

A064579 Inverse permutation to A054082.

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A054083 a(n) = order of n in the permutation A054082 of the natural numbers if this order exists; a(n) = -1 otherwise.

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A054084 Permutation of N: for each k >= 1, let p(k)=least natural number not already an a(i), q(k)=p(k)+k, a(2k-1)=q(k), a(2k)=p(k).

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