Search: id:A056594
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%I A056594
%S A056594 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,
%T A056594 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,
%U A056594 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
%V A056594 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,
%W A056594 -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,
%X A056594 1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0,-1,0,1,0
%N A056594 Periodic sequence 1,0,-1,0...; expansion of 1/(1+x^2).
%C A056594 G.f. is inverse of cyclotomic(4,x). Unsigned: A000035(n+1).
%C A056594 a(n)=1/2((-i)^n + i^n), where i = sqrt(-1). - Mitch Harris, Apr 19 2005.
%C A056594 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
%C A056594 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
%H A056594 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A056594 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A056594 (1/2) [I^n + (-I)^n].
%F A056594 (1/2) {(-1)^(n+Floor(n/2)) + (-1)^Floor(n/2)}.
%F A056594 Recurrence: a(n)=a(n-4), a(0)=1, a(1)=0, a(2)=-1, a(3)=0.
%F A056594 a(n)= cos(n*Pi/2), with n>=0. - Paolo P. Lava (ppl(AT)spl.at), Aug 02 
               2006
%F A056594 G.f.: 1/(1+x^2). E.g.f.: cos(x).
%t A056594 CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
%o A056594 (PARI) {a(n) = real( I^n )}
%o A056594 (PARI) {a(n) = kronecker(-4, n+1) }
%o A056594 sage: [lucas_number1(n,0,1) for n in xrange(1,94)] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Jul 06 2008
%Y A056594 a(n)=S(n, 0)= A049310(n, 0); S(n, x) := U(n, x/2), Chebyshev polynomials 
               of 2nd kind.
%Y A056594 Cf. A049310, A074661, A131852.
%Y A056594 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]
%Y A056594 Sequence in context: A015757 A166698 A101455 this_sequence A091337 A059841 
               A071022
%Y A056594 Adjacent sequences: A056591 A056592 A056593 this_sequence A056595 A056596 
               A056597
%K A056594 easy,sign
%O A056594 0,1
%A A056594 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 04 
               2000

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