A067060 -id:A067060 - 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

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.

A067061 A permutation of the natural numbers.

Original entry on oeis.org

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

Offset: 1

Amarnath Murthy, Jan 03 2002

nonn

Start with the sequence of natural numbers. Rearrange the sequence so that any two consecutive numbers differ by at least 3, by the following process.
Move 1 by the minimum number of steps required to the right.
Move 2 by the minimum number of steps required to the right, etc.
Move the first element which is required to be moved by the minimum number of steps in the sequence obtained by the previous step.
Initial sequence 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,...
after one step.. 2,3,4,1,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,...
after two steps. 3,4,1,5,2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,...
after 3 steps... 4,1,5,2,6,3,8,9,10,7,11,12,13,14,15,16,17,18,19,...
Start with 4. Decrease by 3 then increase by 4 then decrease by 3 and then increase by 4 Decrease by 3 then increase by 7 to obtain first seven terms. Repeat the process for getting the subsequent terms.

Cf. A067060.

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2002

A227792 Expansion of (1 + 6x + 17x^2 - x^3 - 3x^4)/(1 - 6x^2 + x^4).

Original entry on oeis.org

1, 6, 23, 35, 134, 204, 781, 1189, 4552, 6930, 26531, 40391, 154634, 235416, 901273, 1372105, 5253004, 7997214, 30616751, 46611179, 178447502, 271669860, 1040068261, 1583407981, 6061962064, 9228778026, 35331704123, 53789260175, 205928262674

Offset: 0

Ralf Stephan, Sep 23 2013

Also, values i where A067060(i)/i reaches a new maximum (conjectured).

E. I. Emerson, Recurrent Sequences in the Equation DQ^2=R^2+N, Fib. Quart., 7 (1969), pp. 231-242.
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1)

Cf. A041017.

CoefficientList[Series[(1+6x+17x^2-x^3-3x^4)/(1-6x^2+x^4),{x,0,40}],x] (* or *) LinearRecurrence[{0,6,0,-1},{1,6,23,35,134},40] (* Harvey P. Dale, Jun 12 2021 *)

a(n)=polcoeff((-3*x^4-x^3+17*x^2+6*x+1)/(x^4-6*x^2+1)+O(x^100),n)

G.f.: (1+6*x+17*x^2-x^3-3*x^4)/((1+2*x-x^2)*(1-2*x-x^2)).
a(2n) = A038723(n+1), n>0.
a(2n+1) = A001109(n+2).
a(n) = (1/4) * (A135532(n+3) + (-1)^n*A001333(n+2) ).

A115659 Permutation of natural numbers generated by 2-rowed array shown below.

Original entry on oeis.org

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

Offset: 1

Giovanni Teofilatto, Mar 18 2006

0 3 4 7 8 11 12 15 ...: integers congruent to 0 or 3 mod 4.
1 2 5 6 9 10 13 14 ...: integers congruent to 1 or 2 mod 4.
Essentially the same as A067060. - Hugo Pfoertner, Jun 16 2024

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta, UTET, CittaStudiEdizioni, Milano 1997.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A067060, A090964, A116468.

I:=[0, 3, 1, 4, 2, 7]; [n le 6 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Dec 02 2016

Join[{0}, LinearRecurrence[{1, 0, 0, 1, -1}, {3, 1, 4, 2, 7}, 70]] (* Vincenzo Librandi, Dec 02 2016 *)

For n>0, a(n+4k) = a(n) + 4k, with k>=0.

G.f.: x^2*(3 - 2*x + 3*x^2 - 2*x^3 + 2*x^4)/(1 - x - x^4 + x^5). - Philippe Deléham, Dec 02 2016

Entries corrected by R. J. Mathar, Oct 25 2011

cp's OEIS Frontend

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

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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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A067061 A permutation of the natural numbers.

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A227792 Expansion of (1 + 6*x + 17*x^2 - x^3 - 3*x^4)/(1 - 6*x^2 + x^4).

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A115659 Permutation of natural numbers generated by 2-rowed array shown below.

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A227792 Expansion of (1 + 6x + 17x^2 - x^3 - 3x^4)/(1 - 6x^2 + x^4).