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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,... 27
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.

Multiplicative with a(2^e) = 0, a(p^e) = (-1)^((p^e-1)/2) otherwise. - Mitch Harris May 17 2005

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)

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.

Grant Sanderson, Pi hiding in prime regularities (2017)

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (0,-1).

FORMULA

Euler transform of length 4 sequence [ 0, -1, 0, 1]. - 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

G.f.: (x - x^3) / (1 - x^4) = x / (1 + x^2). - Michael Somos, Sep 02 2005

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.

a(n + 1) = A056594(n).

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

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 *)

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

CROSSREFS

Cf. A002654, A009116, A009454, A056594, A108520, A111335, A126120, A146559.

Sequence in context: A015757 A059841 A056594 * A091337 A166698 A179758

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 June 21 19:59 EDT 2018. Contains 305631 sequences. (Running on oeis4.)