cp's OEIS Frontend

A102283 Period 3: repeat [0, 1, -1].

%I A102283 #92 Aug 09 2025 07:47:42
%S A102283 0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,
%T A102283 -1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,
%U A102283 1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1,0,1,-1
%N A102283 Period 3: repeat [0, 1, -1].
%C A102283 The sequence is the non-principal Dirichlet character of the reduced residue system mod 3. (The other is A011655.) Associated Dirichlet L-functions are L(1, chi) = Sum_{n >= 1} a(n)/n = A073010, L(2, chi)= Sum_{n >= 1} a(n)/n^2 = A086724, or L(3, chi)= Sum_{n >= 1} a(n)/n^3 = A129404. [Jolley eq 310] - _R. J. Mathar_, Jul 15 2010
%C A102283 a(n) = 2*D(n) - L(n), where L(n) denotes the n-th Lucas number and D(n) denotes the so-called n-th quadrapell number -- defined and discussed by Dursun Tasci in his paper (see References below). We have D(n) = D(n-2) + 2*D(n-3) + D(n-4), D(0) = D(1) = D(2) = 1, D(3) = 2. G.f. D(x) = (1+x-x^3)/((1-x-x^2)(1+x+x^2)). - _Roman Witula_, Jul 31 2012
%C A102283 This is a strong elliptic divisibility sequence t_n as given in [Kimberling, p. 16] where x = -1, y = 0, z = -1. - _Michael Somos_, Nov 27 2019
%D A102283 M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
%D A102283 L. B. W. Jolley, Summation of Series, Dover Publications (1961).
%D A102283 Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 6.
%H A102283 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/17-1/kimberling1.pdf">Strong divisibility sequences and some conjectures</a>, Fib. Quart., 17 (1979), 13-17.
%H A102283 R. J. Mathar, <a href="http://arxiv.org/abs/1008.2547">Table of Dirichlet L-series..</a>, arXiv:1008.2547 [math.NT], 2010-2015, Table 2, Table 22 for m=3, r=2.
%H A102283 D. Tasci, <a href="http://dergipark.ulakbim.gov.tr/hujms/article/view/5000017451">On Quadrapell Numbers and Quadrapell Polynomials</a>, Hacettepe J. Math. Stat., 38 (3) (2009), 265-275.
%H A102283 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol.</a>
%H A102283 Wikipedia, <a href="https://en.wikipedia.org/wiki/Kronecker_symbol">Kronecker Symbol.</a>
%H A102283 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1).
%F A102283 a(n) = A049347(n-1).
%F A102283 a(n) = -a(n-1) - a(n-2); a(0) = 0, a(1) = 1. G.f.: x/(1+x+x^2). - _Philippe Deléham_, Nov 03 2008
%F A102283 a(n) = -2*sin(4*Pi*n/3)/sqrt(3) = 2*sin(8*Pi*n/3)/sqrt(3). - _Jaume Oliver Lafont_, Dec 05 2008
%F A102283 a(n) = 2*sin(2*Pi*n/3)/sqrt(3). - _Roman Witula_, Jul 31 2012
%F A102283 a(n) = Legendre(n, 3), the Legendre symbol for p = 3. - _Alonso del Arte_, Feb 06 2013
%F A102283 a(n) = (-3/n), where (k/n) is the Kronecker symbol. See the Eric Weisstein and Wikipedia links. - _Wolfdieter Lang_, May 29 2013
%F A102283 Dirichlet g.f.: L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3. - _Ralf Stephan_, Mar 27 2015
%F A102283 a(n) = a(n-3) for n > 2. - _Wesley Ivan Hurt_, Jul 02 2016
%F A102283 E.g.f.: 2*sin(sqrt(3)*x/2)*exp(-x/2)/sqrt(3). - _Ilya Gutkovskiy_, Jul 02 2016
%F A102283 a(n) = H(2*n, 1, 1/2) for n > 0 where H(n, a, b) = hypergeom([a - n/2, b - n/2], [1 - n], 4). - _Peter Luschny_, Sep 03 2019
%F A102283 Euler transform of length 3 sequence [-1, 0, 1]. - _Michael Somos_, Nov 27 2019
%F A102283 a(n) = n - 3*floor((n+1)/3). - _Wolfdieter Lang_, Oct 07 2021
%e A102283 G.f. = x - x^2 + x^4 - x^5 + x^7 - x^8 + x^10 - x^11 + ... - _Michael Somos_, Nov 27 2019
%p A102283 ch:=n-> if n mod 3 = 0 then 0; elif n mod 3 = 1 then 1; else -1; fi;
%p A102283 seq(op([0, 1, -1]), n=1..50); # _Wesley Ivan Hurt_, Jul 02 2016
%t A102283 Table[JacobiSymbol[n, 3], {n, 0, 99}] (* _Alonso del Arte_, Feb 06 2013 *)
%t A102283 Table[KroneckerSymbol[-3, n], {n, 0, 99}] (* _Wolfdieter Lang_, May 30 2013 *)
%t A102283 PadRight[{}, 100, {0, 1, -1}] (* _Wesley Ivan Hurt_, Jul 02 2016 *)
%t A102283 a[ n_] := {1, -1, 0}[[Mod[n, 3, 1]]]; (* _Michael Somos_, Nov 27 2019 *)
%o A102283 (Sage)
%o A102283 def A102283():
%o A102283     x, y = 0, -1
%o A102283     while True:
%o A102283         yield -x
%o A102283         x, y = y, -x -y
%o A102283 a = A102283(); [next(a) for i in range(40)]  # _Peter Luschny_, Jul 11 2013
%o A102283 (Magma) &cat [[0, 1, -1]^^30]; // _Wesley Ivan Hurt_, Jul 02 2016
%o A102283 (PARI) a(n)=([0,1; -1,-1]^n*[0;1])[1,1] \\ _Charles R Greathouse IV_, Jan 14 2017
%o A102283 (PARI) {a(n) = [0, 1, -1][n%3 + 1]}; /* _Michael Somos_, Nov 27 2019 */
%o A102283 (Python)
%o A102283 def A102283(n): return (0,1,-1)[n%3] # _Chai Wah Wu_, Sep 16 2023
%Y A102283 Cf. A011655, A049347, A073010, A086724, A129404, A002324 (Mobius transform).
%K A102283 sign,easy,mult
%O A102283 0,1
%A A102283 _N. J. A. Sloane_, Nov 02 2008