A263449 -id:A263449 - 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

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

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Magma

Magma

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