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a(n) = (2/n), where (k/n) is the Kronecker symbol.
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%I #80 Nov 14 2024 12:23:59

%S 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,

%T 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,

%U -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

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

%C Sinh(1) in 'reflected factorial' base is 1.01010101010101010101010101010101010101010101... see A073097 for cosh(1). - _Robert G. Wilson v_, May 04 2005

%C 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]

%C 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.]

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

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

%H Jianing Song, <a href="/A091337/b091337.txt">Table of n, a(n) for n = 0..1000</a>

%H John M. Campbell, <a href="http://arxiv.org/abs/1105.3399">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv:1105.3399 [math.GM], 2011.

%H R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series...</a>, arXiv:1008.2547 [math.NT], 2010, 2015, L(m=8,r=2,s).

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rfmc.html">Rational Function Multiplicative Coefficients</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).

%F Euler transform of length 8 sequence [0, -1, 0, -1, 0, 0, 0, 1]. - _Michael Somos_, Jul 17 2009

%F 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

%F 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

%F Transform of Pell numbers A000129 by the Riordan array A102587. - _Paul Barry_, Jul 14 2005

%F a(n) = (2/n) = (n/2), _Charles R Greathouse IV_ explained. - _Alonso del Arte_, Oct 31 2014

%F 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

%F From _Jianing Song_, Nov 14 2018: (Start)

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

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

%F Moebius transform of A035185.

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

%F a(n) = Sum_{i=1..n} (-1)^(i + floor((i-3)/4)). - _Wesley Ivan Hurt_, Apr 27 2020

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

%p 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

%t KroneckerSymbol[Range[100], 2] (* _Alonso del Arte_, Oct 30 2014 *)

%o (PARI) {a(n) = (n%2) * (-1)^((n+1)\4)}; /* _Michael Somos_, Sep 10 2005 */

%o (PARI) {a(n) = kronecker( 2, n)}; /* _Michael Somos_, Sep 10 2005 */

%o (PARI) {a(n) = [0, 1, 0, -1, 0, -1, 0, 1][n%8 + 1]}; /* _Michael Somos_, Jul 17 2009 */

%o (Magma) [(n mod 2) * (-1)^((n+1) div 4) : n in [1..100]]; // _Vincenzo Librandi_, Oct 31 2014

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

%Y 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).

%K sign,mult,easy,changed

%O 0,1

%A _Eric W. Weisstein_, Dec 30 2003

%E a(0) prepended by _Jianing Song_, Nov 14 2024