A092486 -id:A092486 - cp's OEIS frontend

A056699 First differences are 2,1,-2,3 (repeated).

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

Offset: 1

Michael Knauth (knauth_jur(AT)yahoo.de), Nov 21 2003

Second quadrisection of natural numbers shifted right two places. - Ralf Stephan, Jun 10 2005

A permutation of the natural numbers partitioned into quadruples [4k-3,4k-1,4k,4k-2] for k > 0. Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the second and third elements, then swap the third and fourth elements; repeat for all quadruples. - Guenther Schrack, Oct 18 2017

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

Inverse: A284307.
Sequence of fixed points: A016813(n-1) for n > 0.
Odd elements: A005408(n-1) for n > 0.
Indices of odd elements: A042963(n) for n > 0.
Even elements: 2*A103889(n) for n > 0.
Indices of even elements: A014601(n) for n > 0.
Cf. A067060, A092486, A116966, A132400, A292576, A263449, A042938 (bisection)

a = [1 3 4 2];
max = 10000;  % Generation of a b-file
for n := 5:max
   a(n) = a(n-4) + 4;
end;
% Guenther Schrack, Oct 18 2017

[Floor((n - ((-1)^n + (-1)^(n*(n-1)/2)*(2+(-1)^n)) / 2)): n in [1..100]]; // Vincenzo Librandi, Feb 05 2018

LinearRecurrence[{1,0,0,1,-1},{1,3,4,2,5},70] (* Harvey P. Dale, May 10 2014 *)
Table[Floor[(n - ((-1)^n + (-1)^(n (n - 1) / 2) (2 + (-1)^n)) / 2)], {n, 100}] (* Vincenzo Librandi, Feb 05 2018 *)

for(n=1, 10000, print1(n - ((-1)^n + (-1)^(n*(n-1)/2)*(2+(-1)^n))/2, ", ")) \\ Guenther Schrack, Oct 18 2017

G.f.: x*(2*x^4 - 2*x^3 + x^2 + 2*x + 1)/((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Nov 08 2012
From Guenther Schrack, Oct 18 2017: (Start)
a(n) = a(n-4) + 4 for n > 4.
a(n) = n + periodic[0,1,1,-2].
a(n) = A092486(A067060(n) - 1) for n > 0.
a(n) = A292576(n) - 2*((-1)^floor(n/2)) for n > 0.
a(A116966(n-1)) = A263449(n-1) for n > 0.
A263449(a(n) - 1) = A116966(n-1) for n > 0.
a(n+2) - a(n) = (-1)^floor(n^2/4)*A132400(n+1) for n > 0.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5. (End)
a(n) = A298364(n-1) + 1 for n > 1. - Guenther Schrack, Feb 04 2018

A159966 Lodumo_4 of A102370 (sloping binary numbers).

Original entry on oeis.org

0, 3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72

Offset: 0

Philippe Deléham, Apr 28 2009

A permutation of the nonnegative integers.
A092486 preceded by a zero. - Philippe Deléham, May 05 2009
Fixed points are the even numbers. - Wesley Ivan Hurt, Oct 16 2015

G. C. Greubel, Table of n, a(n) for n = 0..10000
OEIS wiki, Lodumo transform.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A005843, A102370, A132429, A158459, A166549.

[n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n) div 4) : n in [0..100]]; // Wesley Ivan Hurt, Oct 16 2015

A159966:=n->n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4): seq(A159966(n), n=0..100); # Wesley Ivan Hurt, Oct 16 2015

Table[n - (1 - (-1)^n) (-1)^((2 n + 1 - (-1)^n)/4), {n, 0, 40}] (* or *) CoefficientList[Series[(3 x - 4 x^2 + 3 x^3)/((x - 1)^2 (1 + x^2)), {x, 0, 100}], x] (* Wesley Ivan Hurt, Oct 16 2015 *)
LinearRecurrence[{2,-2,2,-1},{0,3,2,1},80] (* Harvey P. Dale, Jul 02 2022 *)

concat(0, Vec((3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)) + O(x^100))) \\ Altug Alkan, Oct 17 2015

a(n) = lod_4 (A102370(n)).
From Wesley Ivan Hurt, Oct 16 2015: (Start)
G.f.: (3*x-4*x^2+3*x^3)/((x-1)^2*(1+x^2)).
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4), n>3.
a(n) = n-(1-(-1)^n)*(-1)^((2*n+1-(-1)^n)/4).
a(2n) = A005843(n); a(2n+1) = A166549(n).
a(n+1) - a(n) = A132429(n)*(-1)^n. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Mar 24 2017

Partition the natural number sequence into quadruples starting with (1, 2, 3, 4); swap the third and fourth element, then swap the second and third element; repeat for all quadruples.

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

Inverse: A056699.
Subsequences:
  elements with odd index: A042963(n), n > 0
  elements with even index: A014601(A103889(n)), n > 0
  odd elements: A005408(n-1), n > 0
  indices of odd elements: A042948(n), n > 0
  even elements: 2*A103889(n), n > 0
  indices of even elements: A042964(n), n > 0
Sequence of fixed points: A016813(n-1), n > 0
Every fourth element starting at:
  n=1: a(4n-3) = 4n-3 = A016813(n-1), n > 0
  n=2: a(4n-2) = 4n = A008586(n), n > 0
  n=3: a(4n-1) = 4n-2 = A016825(n-1), n > 0
  n=4: a(4n) = 4n-1 = A004767(n-1), n > 0
Difference between pairs of elements:
  a(2n+1)-a(2n-1) = A010684(n-1), n > 0
Compositions:
  a(n) = A133256(A116966(n-1)), n > 0
  a(A042948(n)) = A005408(n-1), n > 0
  A067060(a(n)) = A092486(n), n > 0

a = [1 4 2 3];
max = (specify);
for n = 5:max
   a(n) = a(n-4) + 4;
end;

Table[n + ((-1)^n - (-1)^(n (n - 1)/2) (1 + 2 (-1)^n))/2, {n, 68}] (* Michael De Vlieger, Mar 28 2017 *)
LinearRecurrence[{1,0,0,1,-1},{1,4,2,3,5},70] (* or *) {#[[1]],#[[4]], #[[2]],#[[3]]}&/@Partition[Range[70],4]//Flatten(* Harvey P. Dale, Sep 27 2017 *)

for(n=1, 68, print1(n + ((-1)^n - (-1)^(n*(n - 1)/2)*(1 + 2*(-1)^n))/2,", ")) \\ Indranil Ghosh, Mar 29 2017

a(1)=1, a(2)=4, a(3)=2, a(4)=3, a(n) = a(n-4) + 4, n > 4.
O.g.f.: (x^4 + x^3 - 2*x^2 + 3x - 1)/(x^5 - x^4 - x + 1).
a(n) = n + ((-1)^n - (-1)^(n*(n-1)/2)*(1 + 2*(-1)^n))/2.
a(n) = n + (-1)^n*(1 - (-1)^(n*(n-1)/2) - (i^n - (-i)^n))/2.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5), n > 5.
First differences, periodic: (3, -2, 1, 2), repeat.
a(n) = (2*n - 3*cos(n*Pi/2) + cos(n*Pi) + sin(n*Pi/2))/2. - Wesley Ivan Hurt, Apr 01 2017

A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

Original entry on oeis.org

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

Offset: 1

Guenther Schrack, Jan 18 2018

Partition the natural number sequence into quadruples starting with (1,2,3,4); swap the first and second elements, then swap the second and third elements; repeat for all quadruples.

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

Inverse: A292576.
Sequence of fixed points: A008586(n) for n > 0.
First differences: (-1)^floor(n^2/4)*A068073(n-1) for n > 0.
Subsequences:
  elements with odd index: A042963(A103889(n)) for n > 0.
  elements with even index A014601(n) for n > 0.
  odd elements: A166519(n-1) for n > 0.
  indices of odd elements: A042964(n) for n > 0.
  even elements: A005843(n) for n > 0.
  indices of even elements: A042948(n) for n > 0.
Other similar permutations: A116966, A284307, A292576.

a = [2 3 1 4];
max = 10000;    % Generation of b-file.
for n := 5:max
   a(n) = a(n-4) + 4;
end;

Nest[Append[#, #[[-4]] + 4] &, {2, 3, 1, 4}, 63] (* or *)
Array[# + ((-1)^# + ((-1)^(# (# - 1)/2)) (1 - 2 (-1)^#))/2 &, 67] (* Michael De Vlieger, Jan 23 2018 *)
LinearRecurrence[{1,0,0,1,-1},{2,3,1,4,6},70] (* Harvey P. Dale, Dec 12 2018 *)

for(n=1, 100, print1(n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2, ", "))

O.g.f.: (3*x^3 - 2*x^2 + x + 2)/(x^5 - x^4 - x - 1).
a(1) = 2, a(2) = 3, a(3) = 1, a(4) = 4, a(n) = a(n-4) + 4 for n > 4.
a(n) = n + ((-1)^n + ((-1)^(n*(n-1)/2))*(1 - 2*(-1)^n))/2.
a(n) = n + (cos(n*Pi) - cos(n*Pi/2) + 3*sin(n*Pi/2))/2.
a(n) = 2*floor((n+1)/2) - 4*floor((n+1)/4) + floor(n/2) + 2*floor(n/4).
a(n) = n + (-1)^floor((n-1)^2/4)*A140081(n) for n > 0.
a(n) = A056699(n+1) - 1, n > 0.
a(n+2) = A168269(n+1) - a(n), n > 0.
a(n+2) = a(n) + (-1)^floor((n+1)^2/4)*A132400(n+2) for n > 0.
Linear recurrence: a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
First differences: periodic, (1, -2, 3, 2) repeat.
Compositions:
  a(n) = A080412(A116966(n-1)) for n > 0.
  a(n) = A284307(A256008(n)) for n > 0.
  a(A067060(n)) = A133256(n) for n > 0.
  A116966(a(n+1)-1) = A092486(n) for n >= 0.
  A056699(a(n)) = A256008(n) for n > 0.

A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

Original entry on oeis.org

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

Offset: 0

Wesley Ivan Hurt, Oct 17 2015

Fixed points are the odd numbers (A005408).

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A004442, A005408, A092486, A132429, A176742, A244009, A263449.

[n+(1+(-1)^n)*(-1)^(n*(n+1) div 2) : n in [0..80]];

/* By definition: */ &cat[[4*k+2,4*k+1,4*k,4*k+3]: k in [0..20]]; // Bruno Berselli, Nov 08 2015

A263426:=n->n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2): seq(A263426(n), n=0..80);

Table[n + (1 + (-1)^n)*(-1)^(n*(n + 1)/2), {n, 0, 80}]

Vec((2-3*x+2*x^2+x^3)/((x-1)^2*(1+x^2)) + O(x^100)) \\ Altug Alkan, Oct 19 2015

G.f.: (2 - 3*x + 2*x^2 + x^3)/((x - 1)^2*(1 + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>3.
a(n) = n + (1 + (-1)^n)*(-1)^(n*(n+1)/2).
a(n) = 4*floor((n+1)/4) - (n mod 4) + 2.
a(n) = A092486(n) - 1.
a(n) = n + A176742(n) for n>0.
a(2n) = 2*A004442(n), a(2n+1) = A005408(n).
a(-n-1) = -A263449(n).
a(n+1) = a(n) - A132429(n+1)*(-1)^n.
Sum_{n>=0, n!=2} (-1)^(n+1)/a(n) = 1 - log(2) (A244009). - Amiram Eldar, Dec 25 2023

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A056699 First differences are 2,1,-2,3 (repeated).

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A159966 Lodumo_4 of A102370 (sloping binary numbers).

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A284307 Permutation of the natural numbers partitioned into quadruples [4k-3, 4k, 4k-2, 4k-1], k > 0.

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A298364 Permutation of the natural numbers partitioned into quadruples [4k-2, 4k-1, 4k-3, 4k] for k > 0.

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A263426 Permutation of the nonnegative integers: [4k+2, 4k+1, 4k, 4k+3, ...].

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