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 A033999 a(n) = (-1)^n. 115
 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1 (list; graph; refs; listen; history; text; internal format)
 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 LINKS S. K. Ghosal, J. K. Mandal, Stirling Transform Based Color Image Authentication, Procedia Technology, 2013 Volume 10, 2013, Pages 95-104. Tanya Khovanova, Recursive Sequences Eric Weisstein's World of Mathematics, Inverse Tangent Eric Weisstein's World of Mathematics, Stirling Transform Wikipedia, Grandi's series Wikipedia, +/-1-sequence Wikipedia, Dirichlet eta function 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_{0<=k<=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>1, 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 + ... MAPLE A033999 := n->(-1)^n: seq(A033999(n), n=0..100); 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 CROSSREFS Cf. A056594, A059841, A063007, A122803. Sequence in context: A143622 A076479 A155040 * A000012 A162511 A157895 Adjacent sequences:  A033996 A033997 A033998 * A034000 A034001 A034002 KEYWORD sign,easy AUTHOR Vasiliy Danilov (danilovv(AT)usa.net), Jun 15 1998 STATUS approved

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Last modified August 20 16:49 EDT 2018. Contains 313926 sequences. (Running on oeis4.)