A038608 a(n) = n*(-1)^n.

0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60, -61, 62, -63, 64, -65

Offset: 0

Author

Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

Keywords

• sign
• easy

Comments

a(n) is the determinant of the (n+1) X (n+1) matrix with 0's in the main diagonal and 1's elsewhere. - Franz Vrabec, Dec 01 2007

Sum_{n>0} 1/a(n) = -log(2). - Jaume Oliver Lafont, Feb 24 2009

Pisano period lengths: 1, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, ... (is this A066043?). - R. J. Mathar, Aug 10 2012

a(n) is the determinant of the (n+1) X (n+1) matrix whose i-th row, j-th column entry is the value of the cubic residue symbol ((j-i)/p) where p is a prime of the form 3k+2 and n < p. - Ryan Wood, Nov 09 2017

a(n-1) is the difference in the number of even minus odd parity derangements (permutations with no fixed points) in symmetric group S_n. - Julian Hatfield Iacoponi, Aug 01 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..2000
• Tanya Khovanova, Recursive Sequences
• László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
• Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Crossrefs

Cf. A002162 (log(2)).

Cf. A001477.

Cf. A003221, A000387 (even, odd derangements).

Programs

Haskell

a038608 n = n * (-1) ^ n
a038608_list = [0, -1] ++ map negate
   (zipWith (+) a038608_list (map (* 2) $ tail a038608_list))
-- Reinhard Zumkeller, Nov 24 2012

Magma

[n*(-1)^n: n in [0..80]]; // Vincenzo Librandi, Jun 08 2011

Maple

A038608 := n->n*(-1)^n; seq(A038608(n), n=0..100);

Mathematica

Array[# (-1)^# &, 66, 0] (* Michael De Vlieger, Nov 18 2017 *)
Table[If[EvenQ[n],n,-n],{n,0,70}] (* Harvey P. Dale, Jan 17 2022 *)

PARI

a(n)=n*(-1)^n \\ Charles R Greathouse IV, Dec 07 2011

Python

def A038608(n): return -n if n&1 else n # Chai Wah Wu, Nov 14 2022

Formula

G.f.: -x/(1+x)^2.

E.g.f: -x*exp(-x).

a(n) = -2*a(n-1) - a(n-2) for n >= 2. - Jaume Oliver Lafont, Feb 24 2009

a(n) = A003221(n+1)-A000387(n+1). - Julian Hatfield Iacoponi, Aug 01 2024

Extensions

Edited by Frank Ellermann, Jan 28 2002