A065190 Self-inverse permutation of the positive integers: 1 is fixed, followed by an infinite number of adjacent transpositions (n n+1).

1, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24, 27, 26, 29, 28, 31, 30, 33, 32, 35, 34, 37, 36, 39, 38, 41, 40, 43, 42, 45, 44, 47, 46, 49, 48, 51, 50, 53, 52, 55, 54, 57, 56, 59, 58, 61, 60, 63, 62, 65, 64, 67, 66, 69, 68, 71, 70, 73

Offset: 1

Author

Antti Karttunen, Oct 19 2001

Keywords

• nonn
• easy

Comments

Also, a lexicographically minimal sequence of distinct positive integers such that a(n) is coprime to n. - Ivan Neretin, Apr 18 2015

The larger term of the pair (a(n), a(n+1)) is always odd. Had we started the sequence with a(1) = 0, it would be the lexicographically first sequence with this property if always extented with the smallest integer not yet present. - Eric Angelini, Feb 17 2017

From Yosu Yurramendi, Mar 21 2017: (Start)

This sequence is self-inverse. Except for the fixed point 1, it consists completely of 2-cycles: (2n, 2n+1), n > 0.

A020651(a(n)) = A020650(n), A020650(a(n)) = A020651(n), n > 0.

A245327(a(n)) = A245328(n), A245328(a(n)) = A245327(n), n > 0.

A063946(a(n)) = a(A063946(n)), n > 0.

A054429(a(n)) = a(A054429(n)) = A092569(n), n > 0.

A258996(a(n)) = a(A258996(n)), n > 0.

A258746(a(n)) = a(A258746(n)), n > 0. (End)

From Enrique Navarrete, Nov 13 2017: (Start)

With a(0)=0, and the rest of the sequence appended, a(n) is the smallest positive number not yet in the sequence such that the arithmetic mean of the first n+1 terms a(0), a(1), ..., a(n) is not an integer; i.e., the sequence is 0, 1, 3, 2, 5, 4, 7, 6, 9, 8, ...

Example: for n=5, (0 + 1 + 3 + 2 + 5)/5 is not an integer.

Fixed points are odd numbers >= 3 and also a(n) = n-2 for even n >= 4. (End)

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000
• F. M. Dekking, Permutations of N generated by left-right filling algorithms, arXiv:2001.08915 [math.CO], 2020.
• Index entries for sequences that are permutations of the natural numbers
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A004442, A065190 o A014681 = A065168, A014681 o A065190 = A065164.

Programs

Magma

[1] cat [n+(-1)^n: n in [2..80]]; // Vincenzo Librandi, Apr 18 2015

Maple

[seq(f(j),j=1..120)]; f := (n) -> `if`((n < 2), n,n+((-1)^n));

Mathematica

f[n_] := Rest@ Flatten@ Transpose[{Range[1, n + 1, 2], {1}~Join~Range[2, n, 2]}]; f@ 72 (* Michael De Vlieger, Apr 18 2015 *)
Rest@ CoefficientList[Series[x (x^3 - 2 x^2 + 2 x + 1)/((x - 1)^2*(x + 1)), {x, 0, 72}], x] (* Michael De Vlieger, Feb 17 2017 *)
Join[{1},LinearRecurrence[{1,1,-1},{3,2,5},80]] (* Harvey P. Dale, Feb 24 2021 *)

PARI

{ for (n=1, 1000, if (n>1, a=n + (-1)^n, a=1); write("b065190.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 13 2009

PARI

x='x+O('x^100); Vec(x*(x^3-2*x^2+2*x+1)/((x-1)^2*(x+1))) \\ Altug Alkan, Feb 04 2016

Python

def a(n): return 1 if n<2 else n + (-1)**n # Indranil Ghosh, Mar 22 2017

R

maxrow <- 8 # by choice
a <- c(1,3,2) # If it were c(1,2,3), it would be A000027
  for(m in 1:maxrow) for(k in 0:(2^m-1)){
a[2^(m+1)+    k] = a[2^m+k] + 2^m
a[2^(m+1)+2^m+k] = a[2^m+k] + 2^(m+1)
}
a
# Yosu Yurramendi, Apr 10 2017

Formula

a(1) = 1, a(n) = n+(-1)^n.

From Colin Barker, Feb 18 2013: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) for n>4.

G.f.: x*(x^3 - 2*x^2 + 2*x + 1) / ((x-1)^2*(x+1)). (End)

a(n)^a(n) == 1 (mod n). - Thomas Ordowski, Jan 04 2016

E.g.f.: x*(1+exp(x)) - 1 + exp(-x). - Robert Israel, Feb 04 2016

a(n) = A014681(n-1) + 1. - Michel Marcus, Dec 10 2016

a(1) = 1, for n > 0 a(2*n) = 2*a(a(n)) + 1, a(2*n + 1) = 2*a(a(n)). - Yosu Yurramendi, Dec 12 2020