OFFSET
0,1
COMMENTS
Period 15: repeat [0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1].
Also a(n) = Kronecker(-15, n).
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)).
Note that (Sum_{i=0..14} i*a(i))/(-15) = 2 gives the class number of the imaginary quadratic field Q(sqrt(-15)).
LINKS
Eric Weisstein's World of Mathematics, Kronecker Symbol
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1,-1,0,1,-1).
FORMULA
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.
Completely multiplicative with a(p) = a(p mod 15) for primes p.
a(n) = a(n+15) = -a(-n) for all n in Z.
From Chai Wah Wu, Feb 16 2021: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) - a(n-5) + a(n-7) - a(n-8) for n > 7.
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)
MATHEMATICA
Array[ JacobiSymbol[#, 15] &, 90, 0] (* Robert G. Wilson v, Aug 06 2018 *)
PadRight[{}, 100, {0, 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1}] (* Harvey P. Dale, Feb 20 2023 *)
PROG
(PARI) a(n) = kronecker(n, 15)
(Magma) [KroneckerSymbol(-15, n): n in [0..100]]; // Vincenzo Librandi, Aug 28 2018
CROSSREFS
Cf. A035175 (inverse Moebius transform).
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)).
KEYWORD
sign,easy,mult
AUTHOR
Jianing Song, Aug 05 2018
STATUS
approved