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%I #97 Apr 06 2024 15:05:20
%S 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,
%T -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 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
%N a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...
%C Called X(n) (i.e., Chi(n)) in Hardy and Wright (p. 241), who show that X(n*m) = X(n)*X(m) for all n and m (i.e., X(n) is completely multiplicative) since (n*m - 1)/2 - (n - 1)/2 - (m - 1)/2 = (n - 1)*(m - 1)/2 == 0 (mod 2) when n and m are odd.
%C Same as A056594 but with offset 1.
%C From _R. J. Mathar_, Jul 15 2010: (Start)
%C The sequence is the non-principal Dirichlet character mod 4. (The principal character is A000035.)
%C Associated Dirichlet L-functions are for example L(1,chi) = Sum_{n>=1} a(n)/n = A003881, or L(2,chi) = Sum_{n>=1} a(n)/n^2 = A006752, or L(3,chi) = Sum_{n>=1} a(n)/n^3 = A153071. (End)
%C a(n) is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = 0, y = -1, z is arbitrary. - _Michael Somos_, Nov 27 2019
%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=4, Chi_2(n).
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241.
%H Muniru A Asiru, <a href="/A101455/b101455.txt">Table of n, a(n) for n = 1..1000</a>
%H Étienne Fouvry, Claude Levesque, Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.
%H Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H Grant Sanderson, <a href="https://www.youtube.com/watch?v=NaL_Cb42WyY">Pi hiding in prime regularities</a>, 3Blue1Brown video (2017).
%H <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-1).
%F Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - _Mitch Harris_ May 17 2005
%F Euler transform of length 4 sequence [0, -1, 0, 1]. - _Michael Somos_, Sep 02 2005
%F G.f.: (x - x^3)/(1 - x^4) = x/(1 + x^2). - _Michael Somos_, Sep 02 2005
%F G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) where f(u, v) = v - u^2 * (1 + 2*v). - _Michael Somos_, Aug 04 2011
%F a(n + 4) = a(n), a(n + 2) = a(-n) = -a(n), a(2*n) = 0, a(2*n + 1) = (-1)^n for all n in Z. - _Michael Somos_, Aug 04 2011
%F a(n + 1) = A056594(n). - _Michael Somos_, Jan 13 2014
%F REVERT transform is A126120. STIRLING transform of A009454. BINOMIAL transform is A146559. BINOMIAL transform of A009116. BIN1 transform is A108520. MOBIUS transform of A002654. EULER transform is A111335. - _Michael Somos_, Mar 30 2012
%F Completely multiplicative with a(p) = 2 - (p mod 4). - _Werner Schulte_, Feb 01 2018
%F a(n) = (-(n mod 2))^binomial(n, 2). - _Peter Luschny_, Sep 08 2018
%F a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - _Jianing Song_, Sep 09 2018
%F From _Jianing Song_, Nov 14 2018: (Start)
%F a(n) = ((-4)/n) (or more generally, ((-4^i)/n) for i > 0), where (k/n) is the Kronecker symbol.
%F E.g.f.: sin(x).
%F Dirichlet g.f. is the Dirichlet beta function.
%F a(n) = A091337(n)*A188510(n). (End)
%e G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ...
%p a := n -> `if`(n mod 2=0, 0, (-1)^((n-1)/2)):
%p seq(a(n), n=1..10^3); # _Muniru A Asiru_, Feb 02 2018
%t a[ n_] := {1, 0, -1, 0}[[ Mod[ n, 4, 1]]]; (* _Michael Somos_, Jan 13 2014 *)
%t LinearRecurrence[{0, -1}, {1, 0}, 75] (* _G. C. Greubel_, Aug 23 2018 *)
%o (PARI) {a(n) = if( n%2, (-1)^(n\2))}; /* _Michael Somos_, Sep 02 2005 */
%o (PARI) {a(n) = kronecker( -4, n)}; /* _Michael Somos_, Mar 30 2012 */
%o (GAP) a := [1, 0];; for n in [3..10^2] do a[n] := a[n-2]; od; a; # _Muniru A Asiru_, Feb 02 2018
%o (Magma) m:=75; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // _G. C. Greubel_, Aug 23 2018
%Y Cf. A002654, A009116, A009454, A056594, A091337, A108520, A111335, A126120, A146559, A188510.
%K sign,mult,easy
%O 1,1
%A _Gerald McGarvey_, Jan 20 2005
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