A116966 a(n) = n + {1,2,0,1} according as n == {0,1,2,3} mod 4.

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

Offset: 0

Author

N. J. A. Sloane, Mar 31 2006

Keywords

• nonn
• easy

Comments

In each group of four consecutive numbers, swap 2nd and 3rd terms. - Zak Seidov, Mar 31 2006

First differences of A089781. - Reinhard Zumkeller, Aug 15 2015

From Guenther Schrack, May 31 2017: (Start)

Permutation of the positive integers partitioned into quadruples [4k+1,4k+3,4k+2,4k+4].

Partial sums: A116996. (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Crossrefs

Cf. A115391, A116996.

Cf. A140081, A089781.

Programs

Haskell

a116966 n = a116966_list !! n
a116966_list = zipWith (+) [0..] $ drop 2 a140081_list
-- Reinhard Zumkeller, Aug 15 2015

Magma

/* By definition: */ [ n + [1,2,0,1][1+(n mod 4)]: n in [0..70] ]; // Bruno Berselli, Nov 25 2012

Maple

f:=proc(i) if i mod 4 = 0 then i+1 elif i mod 4 = 1 then i+2 elif i mod 4 = 2 then i else i+1; fi; end;

Mathematica

b := {1, 2, 0, 1}; a[n_] := n + b[[1 + Mod[n, 4]]]; Table[a[n], {n, 0, 60}] (* Stefan Steinerberger, Mar 31 2006 *)
CoefficientList[Series[(2 x^3 - x^2 + 2 x + 1) / ((x - 1)^2 (x + 1) (x^2 + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 19 2013 *)

Maxima

makelist(n+1+(%i^(n*(n-1))-(-1)^n)/2, n, 0, 70); /* Bruno Berselli, Nov 25 2012 */

PARI

Vec((2*x^3-x^2+2*x+1) / ((x-1)^2*(x+1)*(x^2+1)) + O(x^66) ) \\ Joerg Arndt, Apr 30 2013

Formula

a(n) = n+1+(i^(n(n-1))-(-1)^n)/2, where i=sqrt(-1). - Bruno Berselli, Nov 25 2012

G.f.: (2*x^3-x^2+2*x+1) / ((x-1)^2*(x+1)*(x^2+1)). - Colin Barker, Apr 30 2013

a(n) = A140081(n+2) + n. - Reinhard Zumkeller, Aug 15 2015

From Guenther Schrack, May 31 2017: (Start)

a(n) = n + 1 + ((-1)^(n*(n-1)/2) - (-1)^n)/2.

a(n) = a(n-4) + 4, n > 3.

a(n) = a(n-1) + a(n-4) - a(n-5), n > 4. (End)

Sum_{n>=0} (-1)^n/a(n) = Pi/4 + log(2)/2. - Amiram Eldar, Jan 31 2023