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A056594
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Periodic sequence 1,0,-1,0...; expansion of 1/(1+x^2).
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65
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1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller, Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar, Apr 07 2008
a(n-1),n>=1, is the nontrivial Dirichlet character modulo 4, called Chi_2(4;n) (the trivial one is Chi_1(4;n) given by periodic(1,0) = A000035(n)). See the Apostol reference, p. 139, the k=4, phi(k)=2 table. - Wolfdieter Lang, Jun 21 2011.
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REFERENCES
| T. M. Apostopl, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
| G.f.: 1/(1+x^2).
E.g.f.: cos(x).
a(n) = (1/2)*((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005.
a(n) = (1/2)*((-1)^(n+floor(n/2)) + (-1)^floor(n/2)).
Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
Also a(n) = -a(n-2) for n>1; a(n) = A057077(n)*A000035(n+1) = A010892(A001651(n+1)); a(n) = (-(n mod 4)-(n+1 mod 4)+(n+2 mod 4)+(n+3 mod 4))/4 (cf. forms of modular arithmetic of Paolo P. Lava, i.e. see A146094). - Bruno Berselli, Feb 08 2011
a(n)= cos(n*Pi/2), with n>=0. - Paolo P. Lava, Aug 02 2006
a(n)=T(n, 0)=A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. [From Wolfdieter Lang, Aug 21 2009]
a(n)=S(n, 0)= A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Sum_{k>=0} a(k)/(k+1) = Pi/4 [From Jaume Oliver Lafont, Mar 30 2010]
a(n) = Sum_{k, 0<=k<=n} A101950(n,k)*(-1)^k. - DELEHAM Philippe, Feb 10 2012
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MAPLE
| A056594 := n->(1-irem(n, 2))*(-1)^iquo(n, 2); # Peter Luschny, Jul 27 2011
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MATHEMATICA
| CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
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PROG
| (PARI) {a(n) = real( I^n )}
(PARI) {a(n) = kronecker(-4, n+1) }
(MAGMA) &cat[ [1, 0, -1, 0]: n in [0..23] ]; // Bruno Berselli, Feb 08 2011
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CROSSREFS
| Cf. A049310, A074661, A131852.
Sequence in context: A166698 A059841 * A101455 A091337 A179758 A174888
Adjacent sequences: A056591 A056592 A056593 * A056595 A056596 A056597
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KEYWORD
| easy,sign,changed
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 2000
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