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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... 24
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

Table of n, a(n) for n=1..94.

"3Blue1Brown", 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]}, with n>=1 [From 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

EXAMPLE

G.f. = x - x^3 + x^5 - x^7 + x^9 - x^11 + x^13 - x^15 + x^17 - x^19 + x^21 + ...

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 */

CROSSREFS

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

Sequence in context: A166698 A059841 A056594 * A091337 A179758 A174888

Adjacent sequences:  A101452 A101453 A101454 * A101456 A101457 A101458

KEYWORD

easy,sign,mult

AUTHOR

Gerald McGarvey, Jan 20 2005

STATUS

approved

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Last modified June 23 23:18 EDT 2017. Contains 288676 sequences.