A181983 a(n) = (-1)^(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

Offset

0,3

Comments

This is the Lucas U(-2,1) sequence. - R. J. Mathar, Jan 08 2013

Apparently the Mobius transform of A002129. - R. J. Mathar, Jan 08 2013

For n>0, a(n) is also the determinant of the symmetric n X n matrix M defined by M(i,j) = max(i,j) for 1 <= i,j <= n. - Enrique Pérez Herrero, Jan 14 2013

The sums of the terms of this sequence is the divergent series 1 - 2 + 3 - 4 + ... . Euler summed it to 1/4 which was one of the first examples of summing divergent series. - Michael Somos, Jun 05 2013

Links

Table of n, a(n) for n=0..59.

Enrique Pérez Herrero, Max Determinant, 2013.

Wikipedia, 1 - 2 + 3 - 4 + ....

Wikipedia, Lucas sequence.

Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Index entries for Lucas sequences.

Formula

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

E.g.f.: x / exp(x).

a(n) = -a(-n) = -(-1)^n * A001477(n) for all n in Z.

a(n+1) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = Bernoulli(k) for k = 0, 1, ..., n.

A001787(n) = p(0) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 1, ..., n+1. - Michael Somos, Jun 05 2013

Euler transform of length 2 sequence [-2, 2].

Series reversion of g.f. is A000108(n) (Catalan numbers) with a(0)=0.

Series reversion of e.g.f. is A000169. INVERT transform omitting a(0)=0 is A049347. PSUM transform is A001057. BINOMIAL transform is A154955. - Michael Somos, Jun 05 2013

n * a(n) = A162395(n). - Michael Somos, Jun 05 2013

a(n) = - A038608(n). - Reinhard Zumkeller, Mar 20 2013

a(n+2) = a(n) - 2*(-1)^n. - G. C. Greubel, Aug 11 2018

a(n) = - A274922(n) if n>0. - Michael Somos, Sep 24 2019

From Amiram Eldar, Oct 24 2023: (Start)

Multiplicative with a(2^e) = -2^e, and a(p^e) = p^e for an odd prime p.

Dirichlet g.f.: zeta(s-1) * (1-2^(2-s)). (End)

Example

G.f. = x - 2*x^2 + 3*x^3 - 4*x^4 + 5*x^5 - 6*x^6 + 7*x^7 - 8*x^8 + 9*x^9 + ...

Maple

A181983:=n->-(-1)^n * n; seq(A181983(n), n=0..100); # Wesley Ivan Hurt, Feb 26 2014

Mathematica

a[ n_] := -(-1)^n n;
a[ n_] := Sign[n] SeriesCoefficient[ x / (1 + x)^2, {x, 0, Abs @n}];
a[ n_] := Sign[n] (Abs @n)! SeriesCoefficient[ x / Exp[ x], {x, 0, Abs @n}];
CoefficientList[Series[x/(1+x)^2, {x, 0, 60}], x] (* or *) LinearRecurrence[{-2, -1}, {0, 1}, 60] (* or *) Table[If[OddQ[n], n, -n], {n, 0, 60}]  (* Harvey P. Dale, Apr 22 2022 *)

Prog

(PARI) {a(n) = -(-1)^n * n};
(Haskell)
a181983 = negate . a038608
a181983_list = [0, 1] ++ map negate
   (zipWith (+) a181983_list (map (* 2) $ tail a181983_list))
-- Reinhard Zumkeller, Mar 20 2013
(Magma) [(-1)^(n+1)*n: n in [0..30]]; // G. C. Greubel, Aug 11 2018

Crossrefs

Cf. A000108, A000169, A001057, A001477, A001787, A002129, A038608, A049347, A154955, A274922.

Sequence in context: A274055 A167976 A024000 * A274922 A097141 A160356

Adjacent sequences: A181980 A181981 A181982 * A181984 A181985 A181986

Keyword

sign,mult,easy

Author

Michael Somos, Apr 04 2012

Status

approved