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Search: id:A056594
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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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51
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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; listen)
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OFFSET
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0,1
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COMMENT
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G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
a(n)=1/2((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005.
Real part of i^n and imaginary part of i^(n+1), i=sqrt(-1). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jul 22 2007
The BINOMIAL transform generates A009116(n); the inverse BINOMIAL transform generates (-1)^n*A009116(n). - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Apr 07 2008
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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(1/2) [I^n + (-I)^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.
a(n)= cos(n*Pi/2), with n>=0. - Paolo P. Lava (ppl(AT)spl.at), Aug 02 2006
G.f.: 1/(1+x^2). E.g.f.: cos(x).
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MATHEMATICA
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CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
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PROGRAM
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(PARI) {a(n) = real( I^n )}
(PARI) {a(n) = kronecker(-4, n+1) }
sage: [lucas_number1(n, 0, 1) for n in xrange(1, 94)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 06 2008
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CROSSREFS
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a(n)=S(n, 0)= A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials of 2nd kind.
Cf. A049310, A074661, A131852.
a(n)=T(n, 0)= A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 21 2009]
Sequence in context: A015757 A166698 A101455 this_sequence A091337 A059841 A071022
Adjacent sequences: A056591 A056592 A056593 this_sequence A056595 A056596 A056597
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KEYWORD
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easy,sign
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 2000
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