A091337 a(n) = (2/n), where (k/n) is the Kronecker symbol.

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, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1

Offset: 0

Author

Eric W. Weisstein, Dec 30 2003

Keywords

• sign
• mult
• easy

Comments

Sinh(1) in 'reflected factorial' base is 1.01010101010101010101010101010101010101010101... see A073097 for cosh(1). - Robert G. Wilson v, May 04 2005

A non-principal character for the Dirichlet L-series modulo 8, see arXiv:1008.2547 and L-values Sum_{n >= 1} a(n)/n^s in eq (318) by Jolley. - R. J. Mathar, Oct 06 2011 [The other two non-principal characters are A101455 = {(-4/n)} and A188510 = {(-2/n)}. - Jianing Song, Nov 14 2024]

Period 8: repeat [0, 1, 0, -1, 0, -1, 0, 1]. - Wesley Ivan Hurt, Sep 07 2015 [Adapted by Jianing Song, Nov 14 2024 to include a(0) = 0.]

a(n) = (2^(2i+1)/n), where (k/n) is the Kronecker symbol and i >= 0. - A.H.M. Smeets, Jan 23 2018

Examples

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

References

• L. B. W. Jolley, Summation of series, Dover (1961).

Links

• Jianing Song, Table of n, a(n) for n = 0..1000
• John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv:1105.3399 [math.GM], 2011.
• R. J. Mathar, Table of Dirichlet L-series..., arXiv:1008.2547 [math.NT], 2010, 2015, L(m=8,r=2,s).
• Michael Somos, Rational Function Multiplicative Coefficients
• Eric Weisstein's World of Mathematics, Kronecker Symbol
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Crossrefs

Cf. A000129, A035185, A073097, A087960, A102587.

Kronecker symbols {(d/n)} where d is a fundamental discriminant with |d| <= 24: A109017 (d=-24), A011586 (d=-23), A289741 (d=-20), A011585 (d=-19), A316569 (d=-15), A011582 (d=-11), A188510 (d=-8), A175629 (d=-7), A101455 (d=-4), A102283 (d=-3), A080891 (d=5), this sequence (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), A322829 (d=21), A322796 (d=24).

Programs

Magma

[(n mod 2) * (-1)^((n+1) div 4)  : n in [1..100]]; // Vincenzo Librandi, Oct 31 2014

Maple

A091337:= n -> [0, 1, 0, -1, 0, -1, 0, 1][(n mod 8)+1]: seq(A091337(n), n=1..100); # Wesley Ivan Hurt, Sep 07 2015

Mathematica

KroneckerSymbol[Range[100], 2] (* Alonso del Arte, Oct 30 2014 *)

PARI

{a(n) = (n%2) * (-1)^((n+1)\4)}; /* Michael Somos, Sep 10 2005 */

PARI

{a(n) = kronecker( 2, n)}; /* Michael Somos, Sep 10 2005 */

PARI

{a(n) = [0, 1, 0, -1, 0, -1, 0, 1][n%8 + 1]}; /* Michael Somos, Jul 17 2009 */

Formula

Euler transform of length 8 sequence [0, -1, 0, -1, 0, 0, 0, 1]. - Michael Somos, Jul 17 2009

a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1, 7 (mod 8), a(p^e) = (-1)^e if p == 3, 5 (mod 8). - Michael Somos, Jul 17 2009

G.f.: x*(1 - x^2)/(1 + x^4). a(n) = -a(n + 4) = a(-n) for all n in Z. a(2*n) = 0. a(2*n + 1) = A087960(n). - Michael Somos, Apr 10 2011

Transform of Pell numbers A000129 by the Riordan array A102587. - Paul Barry, Jul 14 2005

a(n) = (2/n) = (n/2), Charles R Greathouse IV explained. - Alonso del Arte, Oct 31 2014

a(n) = (1 - (-1)^n)*(-1)^(n/4 - 1/8 - (-1)^n/8 + (-1)^((2*n + 1 - (-1)^n)/4)/4)/2. - Wesley Ivan Hurt, Sep 07 2015

From Jianing Song, Nov 14 2018: (Start)

a(n) = sqrt(2)*sin(Pi*n/2)*sin(Pi*n/4).

E.g.f.: sqrt(2)*cos(x/sqrt(2))*sinh(x/sqrt(2)).

Moebius transform of A035185.

a(n) = A101455(n)*A188510(n). (End)

a(n) = Sum_{i=1..n} (-1)^(i + floor((i-3)/4)). - Wesley Ivan Hurt, Apr 27 2020

Sum_{n>=1} a(n)/n = A196525. Sum_{n>=1} a(n)/n^2 = A328895. Sum_{n>=1} a(n)/n^3 = A329715. Sum_{n>=1} a(n)/n^4 = A346728. - R. J. Mathar, Dec 17 2024

Extensions

a(0) prepended by Jianing Song, Nov 14 2024