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

0,2

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).

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

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 *)

Prog

(Maxima) makelist(n+1+(%i^(n*(n-1))-(-1)^n)/2, n, 0, 70); \\ Bruno Berselli, Nov 25 2012
(Magma) /* By definition: */ [ n + [1, 2, 0, 1][1+(n mod 4)]: n in [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
(Haskell)
a116966 n = a116966_list !! n
a116966_list = zipWith (+) [0..] $ drop 2 a140081_list
-- Reinhard Zumkeller, Aug 15 2015

Crossrefs

Cf. A115391, A116996.

Cf. A140081, A089781.

Sequence in context: A039906 A056011 A117123 * A182187 A334652 A163241

Adjacent sequences: A116963 A116964 A116965 * A116967 A116968 A116969

Keyword

nonn,easy

Author

N. J. A. Sloane, Mar 31 2006

Status

approved