|
|
A101455
|
|
a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,...
|
|
60
|
|
|
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.
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
|
|
|
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
Completely multiplicative with a(p) = 2 - (p mod 4). - Werner Schulte, Feb 01 2018
a(n) = sin(n*Pi/2) = Im(i^n) where i is the imaginary unit. - Jianing Song, Sep 09 2018
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.
|
|
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)):
|
|
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<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x/(1+x^2))); // G. C. Greubel, Aug 23 2018
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,mult,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|