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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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101
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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
This sequence is duplicated in A101455 but with offset 1. - Gary Detlefs, Oct 04 2013
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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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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry and Nikolaos Pantelidis,On pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays, J Algebr Comb 54, 399-423 (2021).
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Dirichlet beta function
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,-1).
Index entries for sequences related to Chebyshev polynomials.
Index to sequences related to inverse of cyclotomic polynomials
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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.
Also a(n) = -a(n-2) for n>1; a(n) = 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, 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. - 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. - Jaume Oliver Lafont, Mar 30 2010
a(n) = Sum_{k=0..n} A101950(n,k)*(-1)^k. - Philippe Deléham, Feb 10 2012
a(n) = (1/2)*(1 + (-1)^n)*(-1)^(n/2). - Bruno Berselli, Mar 13 2012
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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A056594 := n->(1-irem(n, 2))*(-1)^iquo(n, 2); # Peter Luschny, Jul 27 2011
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MATHEMATICA
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CoefficientList[Series[1/(1 + x^2), {x, 0, 50}], x]
a[n_]:= KroneckerSymbol[-4, n+1]; Table[a[n], {n, 0, 93}]. (* Thanks to Jean-François Alcover. - Wolfdieter Lang, May 31 2013 *)
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
(Maxima) A056594(n) := block(
[1, 0, -1, 0][1+mod(n, 4)]
)$ /* R. J. Mathar, Mar 19 2012 */
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CROSSREFS
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Cf. A049310, A074661, A131852, A002654, A146559 (binomial transf).
Sequence in context: A016213 A015757 A059841 * A101455 A091337 A166698
Adjacent sequences: A056591 A056592 A056593 * A056595 A056596 A056597
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KEYWORD
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sign,easy
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AUTHOR
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Wolfdieter Lang, Aug 04 2000
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STATUS
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approved
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