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A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).

%I A322829 #28 Mar 19 2025 10:06:35
%S A322829 0,1,-1,0,1,1,0,0,-1,0,-1,-1,0,-1,0,0,1,1,0,-1,1,0,1,-1,0,1,1,0,0,-1,
%T A322829 0,-1,-1,0,-1,0,0,1,1,0,-1,1,0,1,-1,0,1,1,0,0,-1,0,-1,-1,0,-1,0,0,1,1,
%U A322829 0,-1,1,0,1,-1,0,1,1,0,0,-1,0,-1,-1,0,-1,0,0,1,1,0,-1,1,0
%N A322829 a(n) = Jacobi (or Kronecker) symbol (n/21).
%C A322829 Period 21: repeat [0, 1, -1, 0, 1, 1, 0, 0, -1, 0, -1, -1, 0, -1, 0, 0, 1, 1, 0, -1, 1].
%C A322829 Also a(n) = Kronecker symbol (21/n).
%C A322829 This sequence is one of the three non-principal real Dirichlet characters modulo 21. The other two are Jacobi or Kronecker symbols {(n/63)} (or {(-63/n)}) and {(n/147)} (or {(-147/n)}).
%H A322829 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>
%H A322829 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1,0,-1,0,1,-1,0,1,-1).
%F A322829 a(n) = 1 for n == 1, 4, 5, 16, 17, 20 (mod 21); -1 for n == 2, 8, 10, 11, 13, 19 (mod 21); 0 for n that are not coprime with 21.
%F A322829 Completely multiplicative with a(p) = a(p mod 21) for primes p.
%F A322829 a(n) = A102283(n)*A175629(n).
%F A322829 a(n) = a(n+21) = -a(n) for all n in Z.
%F A322829 From _Chai Wah Wu_, Feb 18 2021: (Start)
%F A322829 a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-6) + a(n-8) - a(n-9) + a(n-11) - a(n-12) for n > 11.
%F A322829 G.f.: -x*(x - 1)*(x + 1)*(x^8 - 2*x^7 + 2*x^6 + 2*x^2 - 2*x + 1)/(x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1). (End)
%t A322829 JacobiSymbol[Range[0, 100], 21] (* _Paolo Xausa_, Mar 19 2025 *)
%o A322829 (PARI) a(n) = kronecker(n, 21)
%Y A322829 Cf. A035203 (inverse Moebius transform).
%Y A322829 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), A091337 (d=8), A110161 (d=12), A011583 (d=13), A011584 (d=17), this sequence (d=21), A322796 (d=24).
%K A322829 sign,easy,mult
%O A322829 0,1
%A A322829 _Jianing Song_, Dec 27 2018