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a(n) = Kronecker Symbol (-5/n), n >= 0.
4

%I #31 Dec 30 2018 15:44:57

%S 0,1,-1,1,1,0,-1,1,-1,1,0,-1,1,-1,-1,0,1,-1,-1,-1,0,1,1,1,-1,0,1,1,1,

%T 1,0,-1,-1,-1,1,0,1,-1,1,-1,0,1,-1,1,-1,0,-1,1,1,1,0,-1,-1,-1,-1,0,-1,

%U -1,-1,-1,0,1,1,1,1,0,1,1,-1,1,0,-1,-1,-1,1,0,-1,-1,1,-1,0,1,-1,1,1,0,-1,1,1,1

%N a(n) = Kronecker Symbol (-5/n), n >= 0.

%C The number of -1's among the four terms following the 0 at a(5*k), for k >= 0, is 1, 2, 3, 3, 1, 0, 3, 2, 2, 1, 4, 4, 0, 1, 3, 3, 1, 1, 3, 4, ...

%C See the Weisstein link, where it is stated that the period length is 0.

%C In general, the sequence {(k/n)} is not periodic if and only if k == 3 (mod 4). - _Jianing Song_, Dec 30 2018

%H Vincenzo Librandi, <a href="/A226162/b226162.txt">Table of n, a(n) for n = 0..1000</a>

%H Jean-Paul Allouche, Leo Goldmakher, <a href="http://arxiv.org/abs/1608.03957">Mock characters and the Kronecker symbol</a>, arXiv:1608.03957 [math.NT], 2016.

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_symbol">Kronecker Symbol</a>

%F Completely multiplicative with a(2) = -1, a(5) = 0, a(p) = 1 if p == 1, 3, 7, 9 (mod 20), a(p) = -1 if p == 11, 13, 17, 19 (mod 20). - _Jianing Song_, Dec 30 2018

%p 0, seq(numtheory:-jacobi(-5, n), n=1..89); # _Peter Luschny_, Dec 30 2018

%t Table[KroneckerSymbol[-5, n],{n,0,89}].

%o (PARI) a(n)=kronecker(-5,n); \\ _Andrew Howroyd_, Jul 23 2018

%Y Cf. A035183 (inverse Moebius transform).

%Y Sequences related to Kronecker symbols that do not form a Dirichlet character: this sequence {(-5/n)}, A034947 {(-1/n)}, A091338 {(3/n)}, A089509 {(7/n)}.

%Y Cf. A080891 (5/n), A100047.

%K sign,mult

%O 0,1

%A _Wolfdieter Lang_, May 29 2013

%E Keyword:mult added by _Andrew Howroyd_, Jul 23 2018