OFFSET
0,1
COMMENTS
(-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
From Bernard 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 (see A237420 for 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).
REFERENCES
J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, Exercice 10, pp. 14-16.
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..10000
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. Also arXiv: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).
FORMULA
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 in A056594, 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
From Ilya Gutkovskiy, Aug 17 2016: (Start)
a(n) = T_n(-1), where T_n(x) are the Chebyshev polynomials of the first kind.
Binomial transform of A122803. (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
EXAMPLE
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 + ...
MATHEMATICA
Table[(-1)^n, {n, 0, 88}] (* Alonso del Arte, Nov 30 2009 *)
PadRight[{}, 89, {1, -1}] (* Arkadiusz Wesolowski, Sep 16 2012 *)
PROG
(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)
def A033999(n): return -1 if n % 2 else 1 # Chai Wah Wu, May 24 2022
CROSSREFS
KEYWORD
sign,easy,changed
AUTHOR
Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998
STATUS
approved