This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A316569 #52 Feb 16 2025 08:33:56
%S A316569 0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,
%T A316569 0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,
%U A316569 0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1,0
%N A316569 a(n) = Jacobi (or Kronecker) symbol (n, 15).
%C A316569 Period 15: repeat [0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1].
%C A316569 Also a(n) = Kronecker(-15, n).
%C A316569 This sequence is one of the three non-principal real Dirichlet characters modulo 15. The other two are Jacobi or Kronecker symbols (n, 45) (or (45, n)) and (n, 75) (or (-75, n)).
%C A316569 Note that (Sum_{i=0..14} i*a(i))/(-15) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-15)).
%H A316569 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>
%H A316569 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,-1,1,-1,0,1,-1).
%F A316569 a(n) = 1 for n == 1, 2, 4, 8 (mod 15); -1 for n == 7, 11, 13, 14 (mod 15); 0 for n that are not coprime with 15.
%F A316569 Completely multiplicative with a(p) = a(p mod 15) for primes p.
%F A316569 a(n) = A102283(n)*A080891(n).
%F A316569 a(n) = a(n+15) = -a(-n) for all n in Z.
%F A316569 From _Chai Wah Wu_, Feb 16 2021: (Start)
%F A316569 a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-5) + a(n-7) - a(n-8) for n > 7.
%F A316569 G.f.: (x^7 - x^5 + 2*x^4 - x^3 + x)/(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1). (End)
%t A316569 Array[ JacobiSymbol[#, 15] &, 90, 0] (* _Robert G. Wilson v_, Aug 06 2018 *)
%t A316569 PadRight[{},100,{0,1,1,0,1,0,0,-1,1,0,0,-1,0,-1,-1}] (* _Harvey P. Dale_, Feb 20 2023 *)
%o A316569 (PARI) a(n) = kronecker(n, 15)
%o A316569 (Magma) [KroneckerSymbol(-15, n): n in [0..100]]; // _Vincenzo Librandi_, Aug 28 2018
%Y A316569 Cf. A035175 (inverse Moebius transform).
%Y A316569 Kronecker symbols: A063524 ((n, 0) or (0, n)), A000012 ((n, 1) or (1, n)), A091337 ((n, 2) or (2, n) or (n, 8) or (8, n)), A102283 ((n, 3) or (-3, n)), A000035 ((n, 4) or (4, n) or (n, 16) or (16, n)), A080891 ((n, 5) or (5, n)), A109017 ((n, 6) or (-6, n)), A175629 ((n, 7) or (-7, n)), A011655 ((n, 9) or (9, n)), A011582 ((n, 11) or (-11, n)), A134667 ((n, 12) or (-12, n)), A011583 ((n, 13) or (13, n)), this sequence ((n, 15) or (-15, n)).
%K A316569 sign,easy,mult
%O A316569 0,1
%A A316569 _Jianing Song_, Aug 05 2018