cp's OEIS Frontend

A080891 Period 5: repeat [0, 1, -1, -1, 1].

%I A080891 #93 Feb 16 2025 08:32:48
%S A080891 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,
%T A080891 -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,
%U A080891 -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,-1,1,0,1,-1,-1,1,0,1,-1,-1,1,0
%N A080891 Period 5: repeat [0, 1, -1, -1, 1].
%C A080891 a(n) = (5/n), where (k/n) is the Kronecker symbol.
%C A080891 L(1;5) (Dirichlet L-series) is the integral from 0 to 1 of the g.f. of a(n+1). Partial sums are A092202. - _Paul Barry_, Apr 01 2005
%C A080891 From _R. J. Mathar_, Jul 15 2010, simplified Jul 27 2010: (Start)
%C A080891 The sequence is the real non-principal Dirichlet character mod 5. (The principal character mod 5 is A011558.)
%C A080891 Associated Dirichlet L-functions are, for example, L(1,chi) = Sum_{n>=1} a(n)/n = A086466 or L(2,chi) = Sum_{n>=1} a(n)/n^2 = 0.7062114... = 4*Pi^2/(25*sqrt(5)). (End)
%C A080891 This sequence {a(n)} appears in the formula 2*exp(2*Pi*n*i/5) = (A(n) + a(n)*phi) + (C(n) + D(n)*phi)*sqrt(2 + phi)*i, with the golden section phi, i = sqrt(-1) and A(n) = A164116(n+5), C(n) = A156174(n+4) and D(n) = A010891(n+3) for n >= 0. See a comment on A164116. - _Wolfdieter Lang_, Feb 26 2014
%C A080891 In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 4}(x). - _Michael Somos_, Sep 04 2015
%D A080891 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=5, Chi_2(n).
%D A080891 H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1962, p. 173.
%H A080891 J. B. Gil and S. Robins, <a href="http://arxiv.org/abs/math/0309244">Hecke operators on rational functions</a>, arXiv:math/0309244 [math.NT], 2003.
%H A080891 J.-P. Serre, <a href="http://dx.doi.org/10.1090/S0273-0979-03-00992-3">On a theorem of Jordan</a>, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 435.
%H A080891 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KroneckerSymbol.html">Kronecker Symbol</a>.
%H A080891 <a href="/index/Di#divseq">Index to divisibility sequences</a>
%H A080891 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-1,-1,-1).
%H A080891 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%F A080891 If n == 0 (mod 5) a(n)=0; if n == 1 or 4 (mod 5) a(n)=1; if n == 2 or 3 (mod 5) a(n)=-1.
%F A080891 G.f.: x*(1-x^2)/(1+x+x^2+x^3+x^4). - _Paul Barry_, Apr 01 2005
%F A080891 G.f.: x * (1 - x) * (1 - x^2) / (1 - x^5). a(n) = a(-n) = a(n+5) for all n in Z. - _Michael Somos_, Jun 17 2005
%F A080891 Euler transform of length 5 sequence [-1, -1, 0, 0, 1]. - _Michael Somos_, Jun 17 2005
%F A080891 Transform of the Fibonacci numbers by the Riordan array A102587. - _Paul Barry_, Jul 14 2005
%F A080891 a(n) = -1 + floor(12002/99999*10^(n+1)) mod 10. - _Hieronymus Fischer_, Jan 04 2013
%F A080891 a(n) = -1 + floor(137/242*3^(n+1)) mod 3. - _Hieronymus Fischer_, Jan 04 2013
%F A080891 |A011558(n)| = |a(n)| = |A100047(n)|. - _Michael Somos_, May 24 2015
%F A080891 a(n) is completely multiplicative with a(p) = Kronecker(5, p). - _Michael Somos_, Jun 17 2015
%F A080891 From _Wesley Ivan Hurt_, Dec 26 2016: (Start)
%F A080891 a(n) = a(n-5) for n > 4.
%F A080891 a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 3.
%F A080891 a(n) = 1 + 2*floor((n-4)/5) - 2*floor((n-2)/5) + floor((n-1)/5) - floor(n/5). (End)
%F A080891 a(n) = 2*(cos(2*n*Pi/5) - cos(4*n*Pi/5))/sqrt(5). - _Wesley Ivan Hurt_, Sep 26 2018
%F A080891 a(n) = a(n-1)*a(n-4) - a(n-2)*a(n-3) for n > 3. - _Nicolas Bělohoubek_, May 21 2024
%F A080891 a(n) = n^2 - 5*floor((n^2+1)/5). - _Aaron J Grech_, Aug 28 2024
%e A080891 G.f. = x - x^2 - x^3 + x^4 + x^6 - x^7 - x^8 + x^9 + x^11 - x^12 - x^13 + ...
%p A080891 A080891 := proc(n) numtheory[jacobi](n,5) ; end proc: seq(A080891(n),n=0..100) ; # _R. J. Mathar_, Jul 29 2010
%t A080891 a[ n_] := Mod[n^2 + 1, 5] - 1; (* _Michael Somos_, May 24 2015 *)
%t A080891 a[ n_] := KroneckerSymbol[ n, 5]; (* _Michael Somos_, May 24 2015 *)
%t A080891 a[ n_] := {1, -1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* _Michael Somos_, May 24 2015 *)
%t A080891 PadRight[{},120,{0,1,-1,-1,1}] (* _Harvey P. Dale_, Nov 30 2023 *)
%o A080891 (PARI) a(n)=kronecker(5,n) /* Also, a(n)=kronecker(n,5) */
%o A080891 (PARI) {a(n) = (n^2 + 1)%5 - 1}; /* _Michael Somos_, Dec 01 2004 */
%o A080891 (MuPAD) numlib::jacobi(n,5)$ n=0..100 // _Zerinvary Lajos_, May 13 2008
%o A080891 (Magma) &cat [[0, 1, -1, -1, 1]^^30]; // _Wesley Ivan Hurt_, Dec 26 2016
%Y A080891 Cf. A011558, A086937, A100047.
%K A080891 sign,mult,easy
%O A080891 0,1
%A A080891 _N. J. A. Sloane_, Sep 23 2003
%E A080891 Name specified by _Wolfdieter Lang_, Feb 26 2014