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A056594
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Period 4: repeat [1,0,-1,0]; expansion of 1/(1 + x^2).
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108
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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
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OFFSET
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0,1
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COMMENTS
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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
a(n-1), n >= 1, is the character of the Dirichlet beta function. - Daniel Forgues, Sep 15 2012
a(n-1), n >= 1, is also the (strongly) multiplicative function h(n) of Theorem 5.12, p. 150, of the Niven-Zuckerman reference. See the formula section. This function h(n) can be employed to count the integer solutions to n = x^2 + y^2. See A002654 for a comment with the formula. - Wolfdieter Lang, Apr 19 2013
For n >= 2 this gives the determinant of the bipartite graph with 2*n nodes and the adjacency matrix A(n) with elements A(n;1,2) = 1 = A(n;n,n-1), and for 1 < i < n A(n;i,i+1) = 1 = A(n;i,i-1), otherwise 0. - Wolfdieter Lang, Jun 25 2023
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
I. S. Gradstein and I. M. Ryshik, Tables of series, products, and integrals, Volume 1, Verlag Harri Deutsch, 1981.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley (1980), p. 150.
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LINKS
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FORMULA
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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.
a(n) = T(n, 0) = A053120(n, 0); T(n, x) Chebyshev polynomials of the first kind. - 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.
a(0) = 1, a(n-1) = 0 if n is even, a(n-1) = Product_{j=1..m} (-1)^(e_j*(p_j-1)/2) if the odd n-1 = p_1^(e_1)*p_2^(e_2)*...*p_m^(e_m) with distinct odd primes p_j, j=1..m. See the function h(n) of Theorem 5.12 of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
a(n) = (-4/(n+1)), n >= 0, where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. Thanks to Wesley Ivan Hurt. - Wolfdieter Lang, May 31 2013
a(n) = R(n,0)/2 with the row polynomials R of A127672. This follows from the product of the zeros of R, and the formula Product_{k=0..n-1} 2*cos((2*k+1)*Pi/(2*n)) = (1 + (-1)^n)*(-1)^(n/2), n >= 1 (see the Gradstein and Ryshik reference, p. 63, 1.396 4., with x = sqrt(-1)). - Wolfdieter Lang, Oct 21 2013
a(n) = Sum_{k=0..n} i^(k*(k+1)), where i=sqrt(-1). - Bruno Berselli, Mar 11 2015
Dirichlet g.f. of a(n) shifted right: L(chi_2(4),s) = beta(s) = (1-2^(-s))*(d.g.f. of A034947), see comments by Lang and Forgues. - Ralf Stephan, Mar 27 2015
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EXAMPLE
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With a(n-1) = h(n) of Niven-Zuckerman: a(62) = h(63) = h(3^2*7^1) = (-1)^(2*1)*(-1)^(1*3) = -1 = h(3)^2*h(7) = a(2)^2*a(6) = (-1)^2*(-1) = -1. - Wolfdieter Lang, Apr 19 2013
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MAPLE
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MATHEMATICA
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CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
CoefficientList[Series[1/Cyclotomic[4, x], {x, 0, 100}], x] (* Vincenzo Librandi, Apr 03 2014 *)
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PROG
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(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
[1, 0, -1, 0][1+mod(n, 4)]
(Python)
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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