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

Author

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

Keywords

• nonn
• easy

Comments

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

Links

• 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).

Crossrefs

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)

Programs

MATLAB

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

Magma

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

Mathematica

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 *)

PARI

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

Formula

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