A065164 -id:A065164 - cp's OEIS frontend

A014681 Fix 0; exchange even and odd numbers.

0, 2, 1, 4, 3, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15, 18, 17, 20, 19, 22, 21, 24, 23, 26, 25, 28, 27, 30, 29, 32, 31, 34, 33, 36, 35, 38, 37, 40, 39, 42, 41, 44, 43, 46, 45, 48, 47, 50, 49, 52, 51, 54, 53, 56, 55, 58, 57, 60, 59, 62, 61, 64, 63, 66, 65, 68, 67, 70

Offset: 0

Mohammad K. Azarian

A self-inverse permutation of the nonnegative numbers.
If we ignore the first term 0, then this can be obtained as: a(n) is the smallest number different from n, not occurring earlier and coprime to n. - Amarnath Murthy, Apr 16 2003 [Corrected by Alois P. Heinz, May 06 2015]
a(0)=0, a(1)=2, then repeatedly subtract 1 and then add 3. - Jon Perry, Aug 12 2014
The biggest term of the pair [a(n), a(n+1)] is always even. This is the lexicographically first sequence with this property starting with a(1) = 0 and always extented with the smallest integer not yet present. - Eric Angelini, Feb 20 2017

Derek Orr, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Composing this permutation with A065190 gives A065164.
Equals 1 + A004442.
Cf. A103889.

Table[n - (-1)^n, {n, 1, 60}]
Join[{0},LinearRecurrence[{1, 1, -1},{2, 1, 4},69]] (* Ray Chandler, Sep 03 2015 *)

a(n)=n - (-1)^n \\ Charles R Greathouse IV, May 06 2015

G.f.: x*(2-x+x^2)/((1-x)*(1-x^2)). - N. J. A. Sloane
a(n) = n - (-1)^n = a(n-1) + a(n-2) - a(n-3) = a(n-2) + 2. - Henry Bottomley, Mar 29 2000
a(0) = 0; a(2m+1) = 2m+2; for m > 0 a(2m) = 2m - 1. - George E. Antoniou, Dec 04 2001
a(n) = n - (-1)^n + 0^n for n >= 0. - Bruno Berselli, Nov 16 2010
E.g.f.: 1 + (x - 1)*cosh(x) + (1 + x)*sinh(x). - Stefano Spezia, Sep 02 2022

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

Original entry on oeis.org

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

Antti Karttunen, Oct 19 2001

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)

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).

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

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

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

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 *)

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

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

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

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

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

A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 1, 6, 6, 5, 6, 2, 8, 8, 6, 3, 8, 4, 10, 10, 7, 8, 1, 10, 6, 12, 12, 8, 5, 10, 2, 12, 8, 14, 14, 9, 10, 3, 12, 4, 14, 10, 16, 16, 10, 7, 12, 1, 14, 6, 16, 12, 18, 18, 11, 12, 5, 14, 2, 16, 8, 18, 14, 20, 20, 12, 9, 14, 3, 16, 4, 18, 10, 20, 16, 22, 22, 13, 14, 7, 16, 1

Offset: 0

Antti Karttunen, Oct 19 2001

Simple periodic site swap permutations of natural numbers.

Row n of the table (starting from n=0) gives a permutation of natural numbers corresponding to the simple, infinite, periodic site swap pattern ...nnnnn...

			Table begins:
1 2 3 4 5 6 7 ...
2 4 1 6 3 8 5 ...
4 6 2 8 1 10 3 ...
6 8 4 10 2 12 1 ...

Joe Buhler and R. L. Graham, Juggling Drops and Descents, Amer. Math. Monthly, 101, (no. 6) 1994, 507 - 519.
Juggling Information Service, Site Swap FAQs

Successive rows and associated site swap sequences, starting from the zeroth row: (A000027, A000004), (A065164, A000012), (A065165, A007395), (A065166, A010701). Cf. also A065171, A065174, A065177. trinv given at A054425.

PerSS_table := (n) -> PerSS((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1, (n-((trinv(n)*(trinv(n)-1))/2))); PerSS := (n,c) -> Z2N(N2Z(n)+c);
N2Z := n -> ((-1)^n)*floor(n/2); Z2N := z -> 2*abs(z)+`if`((z < 1),1,0);
[seq(PerSS_table(j),j=0..119)];

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then the n-th term of the k-th row is f(g(n)+k).

A065168 Permutation t->t-1 of Z, folded to N.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Oct 19 2001

This permutation consists of just one cycle, which is infinite.

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Inverse permutation to A065164.

Obtained by composing permutations A065190 and A014681.

a:= n-> n-2*(-1)^n +`if`(n=2, 1, 0):
seq(a(n), n=1..80); # Alois P. Heinz, Mar 07 2012

Join[{3, 1}, LinearRecurrence[{1, 1, -1}, {5, 2, 7}, 100]] (* Jean-François Alcover, Feb 28 2016 *)

Let f: Z -> N be given by f(z) = 2z if z>0 else 2|z|+1, with inverse g(z) = z/2 if z even else (1-z)/2. Then a(n) = f(g(n)-1).
G.f.: x*(3-2*x+x^4+x^2-x^3) / ((x+1)*(x-1)^2). - Alois P. Heinz, Mar 07 2012
Sum_{n>=1} (-1)^n/a(n) = 2 - log(2). - Amiram Eldar, Aug 08 2023

A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

Original entry on oeis.org

1, 5, 3, 9, 7, 13, 11, 17, 15, 21, 19, 25, 23, 29, 27, 33, 31, 37, 35, 41, 39, 45, 43, 49, 47, 53, 51, 57, 55, 61, 59, 65, 63, 69, 67, 73, 71, 77, 75, 81, 79, 85, 83, 89, 87, 93, 91, 97, 95, 101, 99, 105, 103, 109, 107, 113, 111, 117, 115, 121, 119

Offset: 0

Paul Barry, Jul 26 2003

Partial sums are A086955.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A004442, A014681, A065190.

[1] cat [2*n+1-2*(-1)^n: n in [1..70]]; // Vincenzo Librandi, Jun 21 2017

Join[{1}, LinearRecurrence[{1, 1, -1}, {5, 3, 9}, 60]] (* Vincenzo Librandi, Jun 21 2017 *)

Vec((1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2) + O(x^100)) \\ Michel Marcus, Jun 21 2017

G.f.: (1+4*x-3*x^2+2*x^3)/((1+x)*(1-x)^2).
a(n) = n + abs(2 - (n + 1)*(-1)^n). - Lechoslaw Ratajczak, Dec 09 2016
a(n) = 2*A065190(n+1)-1 and a(n) = 2*A014681(n)+1. - Michel Marcus, Dec 10 2016
From Guenther Schrack, Jun 09 2017: (Start)
a(n) = 2*n + 1 - 2*(-1)^n for n > 0.
a(n) = 2*n + 1 - 2*cos(n*Pi) for n > 0.
a(n) = 4*n - a(n-1) for n > 1.
Linear recurrence: a(n) = a(n-1) + a(n-2) - a(n-3) for n > 3.
First differences: 2 - 4*(-1)^n for n > 1; -(-1)^n*A010696(n) for n > 1.
a(n) = A065164(n+1) + n for n > 0.
a(A014681(n)) = A005408(n) for n >= 0.
a(A005408(A014681(n)) for n >= 0.
a(n) = A005408(A103889(n)) for n >= 0.
A103889(a(n)) = 2*A065190(n+1) for n >= 0.
a(2*n-1) = A004766(n) for n > 0.
a(2*n+2) = A004767(n) for n >= 0. (End)

A227832 Sum of odd numbers starting with 1, alternating signs (beginning with plus).

Original entry on oeis.org

1, 4, -1, 6, -3, 8, -5, 10, -7, 12, -9, 14, -11, 16, -13, 18, -15, 20, -17, 22, -19, 24, -21, 26, -23, 28, -25, 30, -27, 32, -29, 34, -31, 36, -33, 38, -35, 40, -37, 42, -39, 44, -41, 46, -43, 48, -45, 50, -47, 52, -49

Offset: 1

D.Wilde, Aug 02 2013

1st,3rd,5th (odd terms) increase by 2, 2nd,4th,6th,8th (even terms) decrease by 2 each time.

			(1+3)=4 (4-5)=-1 (-1+7)=6 (6-9)=-3 (-3+11)=8 (8-13)=-5 (-5+15)=10.

Index entries for linear recurrences with constant coefficients, signature (-1,1,1).

Cf. A065164 (absolute values).

a(n) = n*(-1)^n + 2; \\ Michel Marcus, Feb 08 2016

a(n) = -a(n-1) + a(n-2) + a(n-3). - Charles R Greathouse IV, Aug 02 2013

G.f.: x*(2*x^2 + 5*x + 1)/((1-x)*(1+x)^2). a(n) = n*(-1)^n + 2 = A038608(n) + 2. - Ralf Stephan, Aug 07 2013

cp's OEIS Frontend

A014681 Fix 0; exchange even and odd numbers.

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

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A065167 Table T(n,k) read by antidiagonals, where the k-th row gives the permutation t->t+k of Z, folded to N (k >= 0, n >= 1).

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A065168 Permutation t->t-1 of Z, folded to N.

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A086970 Fix 1, then exchange the subsequent odd numbers in pairs.

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A227832 Sum of odd numbers starting with 1, alternating signs (beginning with plus).

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