0,1

(-1)^(n+1) = signed area of parallelogram with vertices (0,0), U=(F(n),F(n+1)), V=(F(n+1),F(n+2)), where F =A000045(Fibonacci numbers). The area of every such parallelogram is 1. The signed area is -1 if and only if F(n+1)^2 > F(n)*F(n+2), or, equivalently, n is even, or, equivalently, the vector U is "above" V, indicating that U and V "cross" as n -> n+1. -Clark Kimberling,Sep 09 2013

Periodic with period length 2. -Ray Chandler,Apr 03 2017

FromBernard Schott,May 11 2022: (Start)

Cesàro mean theorem: When a(n) has a limit (finite or infinite) in the usual sense, then c(n) = (a(1)+...+a(n))/n has the same Cesàro limit, but the converse is false. This sequence is a counterexample in the case of a finite Cesàro limit (seeA237420for counterexample with an infinite Cesàro limit).

This sequence is not convergent in the usual sense because a(2n) = 1 while a(2n+1) = -1; the successive arithmetic means c(n) of the first n terms of the sequence are 1/1, 0/2, 1/3, 0/4, 1/5, 0/6,... so c(2n) = 1/(2n+1) and c(2n+1) = 0, hence the Cesàro limit is 0 because c(n) -> 0 when n -> oo.

In fact, when sequence a(n) is "Period k: [a1, a2,..., ak]", then the Cesàro limit c of this sequence is (a1+a2+...+ak)/k.

Note that the converse of the theorem is true iff a(n) is monotonic (End).

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.

Paul Barry,Characterizations of the Borel triangle and Borel polynomials,arXiv:2001.08799 [math.CO], 2020.

F. Javier de Vega,An extension of Furstenberg's theorem of the infinitude of primes,arXiv:2003.13378 [math.NT], 2020.

S. K. Ghosal and J. K. Mandal,Stirling Transform Based Color Image Authentication,Procedia Technology, 2013 Volume 10, 2013, Pages 95-104.

Tanya Khovanova,Recursive Sequences

Mathematics Stack Exchange,Convergence of series implies convergence of Cesàro mean,2013.

László Németh,The trinomial transform triangle,J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. AlsoarXiv:1807.07109[math.NT], 2018.

ProofWiki,Cesàro mean.

Michael Somos,Rational Function Multiplicative Coefficients

Eric Weisstein's World of Mathematics,Inverse Tangent

Eric Weisstein's World of Mathematics,Stirling Transform

Wikipedia,Ernesto Cesàro.

Wikipedia,Grandi's series

Wikipedia,+/-1-sequence

Wikipedia,Dirichlet eta function

Wikipédia,Lemme de Cesàro(in French).

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

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

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

Linear recurrence: a(0)=1, a(n)=-a(n-1) for n>0. -Jaume Oliver Lafont,Mar 20 2009

Sum_{k=0..n} a(k) =A059841(n). -Jaume Oliver Lafont,Nov 21 2009

Sum_{k>=0} a(k)/(k+1) = log(2). -Jaume Oliver Lafont,Mar 30 2010

Euler transform of length 2 sequence [ -1, 1]. -Michael Somos,Mar 21 2011

Moebius transform is length 2 sequence [ -1, 2]. -Michael Somos,Mar 21 2011

a(n) = -b(n) where b(n) = multiplicative with b(2^e) = -1 if e>0, b(p^e) = 1 if p>2. -Michael Somos,Mar 21 2011

a(n) = a(-n) = a(n + 2) = cos(n * Pi). a(n) = c_2(n) if n>1 where c_k(n) is Ramanujan's sum. -Michael Somos,Mar 21 2011

a(n) = (1/2)*Product_{k=0..2*n-1} 2*cos((2*k+1)*Pi/(4*n)), n >= 1. See the product given in the Oct 21 2013 formula comment inA056594,and replace there n -> 2*n. -Wolfdieter Lang,Oct 23 2013

D.g.f.: (2^(1-s)-1)*zeta(s) = -eta(s) (the Dirichlet eta function). -Ralf Stephan,Mar 27 2015

FromIlya Gutkovskiy,Aug 17 2016: (Start)

a(n) = T_n(-1), where T_n(x) are the Chebyshev polynomials of the first kind.

Binomial transform ofA122803.(End)

a(n) = exp(i*Pi*n) = exp(-i*Pi*n). -Carauleanu Marc,Sep 15 2016

a(n) = Sum_{k=0..n} (-1)^k*A063007(n, k), n >= 0. -Wolfdieter Lang,Sep 13 2016

G.f. = 1 - x + x^2 - x^3 + x^4 - x^5 + x^6 - x^7 + x^8 - x^9 + x^10 - x^11 + x^12 +...

A033999:= n->(-1)^n: seq(A033999(n), n=0..100);

Table[(-1)^n, {n, 0, 88}] (*Alonso del Arte,Nov 30 2009 *)

PadRight[{}, 89, {1, -1}] (*Arkadiusz Wesolowski,Sep 16 2012 *)

(PARI) a(n)=1-2*(n%2) /*Jaume Oliver Lafont,Mar 20 2009 */

(Haskell)

a033999 = (1 -). (* 2). (`mod` 2)

a033999_list = cycle [1, -1] --Reinhard Zumkeller,May 06 2012, Jan 02 2012

(Magma) [(-1)^n: n in [0..100]]; //Wesley Ivan Hurt,Nov 19 2014

(Python)

defA033999(n): return -1 if n % 2 else 1 #Chai Wah Wu,May 24 2022

Cf.A056594,A059841,A063007,A122803.

About Cesàro mean theorem:A114112,A237420.

Cf.A072691(abs. val. Dgf at s=2),A197070(abs. val. Dgf at s=3),A267315(abs. val. Dgf at s=4).

Sequence in context:A076479A155040A209661*A000012A216430A232544

Adjacent sequences:A033996A033997A033998*A034000A034001A034002

sign,easy

Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998

approved