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%I A091337 #83 Feb 16 2025 08:32:52
%S A091337 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 A091337 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 A091337 -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 A091337 a(n) = (2/n), where (k/n) is the Kronecker symbol.
%C A091337 Sinh(1) in 'reflected factorial' base is 1.01010101010101010101010101010101010101010101... see A073097 for cosh(1). - _Robert G. Wilson v_, May 04 2005
%C A091337 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 A091337 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 A091337 a(n) = (2^(2i+1)/n), where (k/n) is the Kronecker symbol and i >= 0. - _A.H.M. Smeets_, Jan 23 2018
%D A091337 L. B. W. Jolley, Summation of series, Dover (1961).
%H A091337 Jianing Song, <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL0EwOTEzMzcvYjA5MTMzNy50eHQ">Table of n, a(n) for n = 0..1000</a>
%H A091337 John M. Campbell, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMTEwNS4zMzk5">An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences</a>, arXiv:1105.3399 [math.GM], 2011.
%H A091337 R. J. Mathar, <a href="/load/view.php?a=aHR0cDovL2FyeGl2Lm9yZy9hYnMvMTAwOC4yNTQ3">Table of Dirichlet L-series...</a>, arXiv:1008.2547 [math.NT], 2010, 2015, L(m=8,r=2,s).
%H A091337 Michael Somos, <a href="/load/view.php?a=aHR0cDovL2dyYWlsLmVlY3MuY3N1b2hpby5lZHUvfnNvbW9zL3JmbWMuaHRtbA">Rational Function Multiplicative Coefficients</a>
%H A091337 Eric Weisstein's World of Mathematics, <a href="/load/view.php?a=aHR0cHM6Ly9tYXRod29ybGQud29sZnJhbS5jb20vS3JvbmVja2VyU3ltYm9sLmh0bWw">Kronecker Symbol</a>
%H A091337 <a href="/load/view.php?a=aHR0cDovL29laXMub3JnL2luZGV4L1JlYyNvcmRlcl8wNA">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,-1).
%F A091337 Euler transform of length 8 sequence [0, -1, 0, -1, 0, 0, 0, 1]. - _Michael Somos_, Jul 17 2009
%F A091337 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 A091337 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 A091337 Transform of Pell numbers A000129 by the Riordan array A102587. - _Paul Barry_, Jul 14 2005
%F A091337 a(n) = (2/n) = (n/2), _Charles R Greathouse IV_ explained. - _Alonso del Arte_, Oct 31 2014
%F A091337 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 A091337 From _Jianing Song_, Nov 14 2018: (Start)
%F A091337 a(n) = sqrt(2)*sin(Pi*n/2)*sin(Pi*n/4).
%F A091337 E.g.f.: sqrt(2)*cos(x/sqrt(2))*sinh(x/sqrt(2)).
%F A091337 Moebius transform of A035185.
%F A091337 a(n) = A101455(n)*A188510(n). (End)
%F A091337 a(n) = Sum_{i=1..n} (-1)^(i + floor((i-3)/4)). - _Wesley Ivan Hurt_, Apr 27 2020
%F A091337 Sum_{n>=1} a(n)/n = A0196525. 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
%e A091337 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 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 A091337 KroneckerSymbol[Range[100], 2] (* _Alonso del Arte_, Oct 30 2014 *)
%o A091337 (PARI) {a(n) = (n%2) * (-1)^((n+1)\4)}; /* _Michael Somos_, Sep 10 2005 */
%o A091337 (PARI) {a(n) = kronecker( 2, n)}; /* _Michael Somos_, Sep 10 2005 */
%o A091337 (PARI) {a(n) = [0, 1, 0, -1, 0, -1, 0, 1][n%8 + 1]}; /* _Michael Somos_, Jul 17 2009 */
%o A091337 (Magma) [(n mod 2) * (-1)^((n+1) div 4)  : n in [1..100]]; // _Vincenzo Librandi_, Oct 31 2014
%Y A091337 Cf. A000129, A035185, A073097, A087960, A102587.
%Y A091337 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 A091337 sign,mult,easy
%O A091337 0,1
%A A091337 _Eric W. Weisstein_, Dec 30 2003
%E A091337 a(0) prepended by _Jianing Song_, Nov 14 2024

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