A130472 A permutation of the integers: a(n) = (-1)^n * floor( (n+1)/2 ).

0, -1, 1, -2, 2, -3, 3, -4, 4, -5, 5, -6, 6, -7, 7, -8, 8, -9, 9, -10, 10, -11, 11, -12, 12, -13, 13, -14, 14, -15, 15, -16, 16, -17, 17, -18, 18, -19, 19, -20, 20, -21, 21, -22, 22, -23, 23, -24, 24, -25, 25, -26, 26, -27, 27, -28, 28, -29, 29, -30, 30, -31, 31, -32, 32

Offset: 0

Author

Clark Kimberling, May 28 2007

Keywords

• sign
• easy

Comments

Pisano period lengths: 1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, ... - R. J. Mathar, Aug 10 2012

Partial sums of A038608. - Stanislav Sykora, Nov 27 2013

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
• Index entries for sequences that are permutations of the integers

Crossrefs

Sums of the form Sum_{k=0..n} k^p * q^k: A059841 (p=0,q=-1), this sequence (p=1,q=-1), A089594 (p=2,q=-1), A232599 (p=3,q=-1), A126646 (p=0,q=2), A036799 (p=1,q=2), A036800 (p=2,q=2), A036827 (p=3,q=2), A077925 (p=0,q=-2), A232600 (p=1,q=-2), A232601 (p=2,q=-2), A232602 (p=3,q=-2), A232603 (p=2,q=-1/2), A232604 (p=3,q=-1/2).

Cf. A001057, A038608.

Programs

Magma

[((-1)^n*(2*n+1)-1)/4: n in [0..80]]; // Vincenzo Librandi, Mar 29 2015

Maple

A130472:=n->ceil(n/2)*(-1)^n; seq(A130472(k), k=0..50); # Wesley Ivan Hurt, Oct 22 2013

Mathematica

CoefficientList[Series[(x^2-x)/(1-x^2)^2, {x, 0, 64}], x] (* Geoffrey Critzer, Sep 29 2013 *)
LinearRecurrence[{-1,1,1},{0,-1,1},70] (* Harvey P. Dale, Mar 02 2018 *)

PARI

a(n)=(-1)^n*n\2 \\ Charles R Greathouse IV, Sep 02 2015

Sage

[((-1)^n*(2*n+1) - 1)/4 for n in (0..70)] # G. C. Greubel, Mar 31 2021

Formula

a(n) = -A001057(n).

a(2n) = n, a(2n+1) = -(n+1).

a(n) = Sum_{k=0..n} k*(-1)^k.

a(n) = -a(n-1) +a(n-2) +a(n-3).

G.f.: -x/( (1-x)*(1+x)^2 ). - R. J. Mathar, Feb 20 2011

a(n) = floor( (n/2)*(-1)^n ). - Wesley Ivan Hurt, Jun 14 2013

a(n) = ceiling( n/2 )*(-1)^n. - Wesley Ivan Hurt, Oct 22 2013

a(n) = ((-1)^n*(2*n+1) - 1)/4. - Adriano Caroli, Mar 28 2015

E.g.f.: (1/4)*(-exp(x) + (1-2*x)*exp(-x) ). - G. C. Greubel, Mar 31 2021