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 A101455 a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,... 41
 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; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. Same as A056594 but with offset 1. From R. J. Mathar, Jul 15 2010: (Start) The sequence is the non-principal Dirichlet character mod 4. (The principal character is A000035.) 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) 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 REFERENCES T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=4, Chi_2(n). G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Oxford Univ. Press, 1979, p. 241. LINKS Muniru A Asiru, Table of n, a(n) for n = 1..1000 Etienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017. C. Kimberling, Strong divisibility sequences and some conjectures, Fib. Quart., 17 (1979), 13-17. Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017) Index entries for linear recurrences with constant coefficients, signature (0,-1). FORMULA Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - Mitch Harris May 17 2005 Euler transform of length 4 sequence [0, -1, 0, 1]. - Michael Somos, Sep 02 2005 G.f.: (x - x^3)/(1 - x^4) = x/(1 + x^2). - Michael Somos, Sep 02 2005 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 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 a(n + 1) = A056594(n). - Michael Somos, Jan 13 2014 a(n) = sin(2*Pi*(n-1))/(4*cos(Pi/2*(n-1))) with n >= 0. - Paolo P. Lava, Jun 20 2006 a(n) = -(1/4)*((n mod 4) - ((n + 1) mod 4) - ((n + 2) mod 4) + ((n + 3) mod 4)). - Paolo P. Lava, Aug 28 2009 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 Completely multiplicative with a(p) = 2 - (p mod 4). - Werner Schulte, Feb 01 2018 a(n) = (-(n mod 2))^binomial(n, 2). - Peter Luschny, Sep 08 2018 a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - Jianing Song, Sep 09 2018 From Jianing Song, Nov 14 2018: (Start) a(n) = ((-4)/n) (or more generally, ((-4^i)/n) for i > 0), where (k/n) is the Kronecker symbol. E.g.f.: sin(x). Dirichlet g.f. is the Dirichlet beta function. a(n) = A091337(n)*A188510(n). (End) EXAMPLE G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ... MAPLE a := n -> `if`(n mod 2=0, 0, (-1)^((n-1)/2)): seq(a(n), n=1..10^3); # Muniru A Asiru, Feb 02 2018 MATHEMATICA a[ n_] := {1, 0, -1, 0}[[ Mod[ n, 4, 1]]]; (* Michael Somos, Jan 13 2014 *) LinearRecurrence[{0, -1}, {1, 0}, 75] (* G. C. Greubel, Aug 23 2018 *) PROG (PARI) {a(n) = if( n%2, (-1)^(n\2))}; /* Michael Somos, Sep 02 2005 */ (PARI) {a(n) = kronecker( -4, n)}; /* Michael Somos, Mar 30 2012 */ (GAP) a := [1, 0];; for n in [3..10^2] do a[n] := a[n-2]; od; a; # Muniru A Asiru, Feb 02 2018 (MAGMA) m:=75; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // G. C. Greubel, Aug 23 2018 CROSSREFS Cf. A002654, A009116, A009454, A056594, A091337, A108520, A111335, A126120, A146559, A188510. Sequence in context: A015757 A059841 A056594 * A091337 A166698 A250299 Adjacent sequences:  A101452 A101453 A101454 * A101456 A101457 A101458 KEYWORD sign,mult,easy AUTHOR Gerald McGarvey, Jan 20 2005 STATUS approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)