A080891 Period 5: repeat [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, 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, 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

Offset

0,1

Comments

a(n) = (5/n), where (k/n) is the Kronecker symbol.

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

From R. J. Mathar, Jul 15 2010, simplified Jul 27 2010: (Start)

The sequence is the real non-principal Dirichlet character mod 5. (The principal character mod 5 is A011558.)

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)

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

In Gil and Robins 2003 on page 33 the g.f. is denoted by f_{4, 4}(x). - Michael Somos, Sep 04 2015

After the Riemann zeta function, the analytic conductor of this L-function is the smallest among L-functions of degree 1. [LMFDB] - Michael Somos, Sep 23 2025

The Dirichlet character associated with the real quadratic field Q(sqrt(5)). - Jianing Song, Dec 13 2025

References

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986, page 139, k=5, Chi_2(n).

H. Cohn, Advanced Number Theory, Dover Publications, Inc., 1962, p. 173.

Links

Table of n, a(n) for n=0..100.

J. B. Gil and S. Robins, Hecke operators on rational functions, arXiv:math/0309244 [math.NT], 2003.

LMFDB, L-function 1-5-5.4-r0-0-0.

J.-P. Serre, On a theorem of Jordan, Bull. Amer. Math. Soc., 40 (No. 4, 2003), 429-440, see p. 435.

Eric Weisstein's World of Mathematics, Kronecker Symbol.

Index to divisibility sequences

Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,-1).

Index entries for two-way infinite sequences

Formula

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.

G.f.: x*(1-x^2)/(1+x+x^2+x^3+x^4). - Paul Barry, Apr 01 2005

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

Euler transform of length 5 sequence [-1, -1, 0, 0, 1]. - Michael Somos, Jun 17 2005

Transform of the Fibonacci numbers by the Riordan array A102587. - Paul Barry, Jul 14 2005

a(n) = -1 + floor(12002/99999*10^(n+1)) mod 10. - Hieronymus Fischer, Jan 04 2013

a(n) = -1 + floor(137/242*3^(n+1)) mod 3. - Hieronymus Fischer, Jan 04 2013

|A011558(n)| = |a(n)| = |A100047(n)|. - Michael Somos, May 24 2015

a(n) is completely multiplicative with a(p) = Kronecker(5, p). - Michael Somos, Jun 17 2015

From Wesley Ivan Hurt, Dec 26 2016: (Start)

a(n) = a(n-5) for n > 4.

a(n) + a(n-1) + a(n-2) + a(n-3) + a(n-4) = 0 for n > 3.

a(n) = 1 + 2*floor((n-4)/5) - 2*floor((n-2)/5) + floor((n-1)/5) - floor(n/5). (End)

a(n) = 2*(cos(2*n*Pi/5) - cos(4*n*Pi/5))/sqrt(5). - Wesley Ivan Hurt, Sep 26 2018

a(n) = a(n-1)*a(n-4) - a(n-2)*a(n-3) for n > 3. - Nicolas Bělohoubek, May 21 2024

a(n) = n^2 - 5*floor((n^2+1)/5). - Aaron J Grech, Aug 28 2024

Example

G.f. = x - x^2 - x^3 + x^4 + x^6 - x^7 - x^8 + x^9 + x^11 - x^12 - x^13 + ...
Dirichlet series L(s) = 1 - 2^-s - 3^-s + 4^-s + 6^-s - 7^-s - 8^-s + 9^-s + ...

Maple

A080891 := proc(n) numtheory[jacobi](n, 5) ; end proc: seq(A080891(n), n=0..100) ; # R. J. Mathar, Jul 29 2010

Mathematica

a[ n_] := Mod[n^2 + 1, 5] - 1; (* Michael Somos, May 24 2015 *)
a[ n_] := KroneckerSymbol[ n, 5]; (* Michael Somos, May 24 2015 *)
a[ n_] := {1, -1, -1, 1, 0}[[Mod[n, 5, 1]]]; (* Michael Somos, May 24 2015 *)
PadRight[{}, 120, {0, 1, -1, -1, 1}] (* Harvey P. Dale, Nov 30 2023 *)

Prog

(PARI) a(n)=kronecker(5, n) /* Also, a(n)=kronecker(n, 5) */
(PARI) {a(n) = (n^2 + 1)%5 - 1}; /* Michael Somos, Dec 01 2004 */
(MuPAD) numlib::jacobi(n, 5)$ n=0..100 // Zerinvary Lajos, May 13 2008
(Magma) &cat [[0, 1, -1, -1, 1]^^30]; // Wesley Ivan Hurt, Dec 26 2016

Crossrefs

Cf. A011558, A086937, A100047.

Moebius transform of A035187.

Cf. A038872 (primes not inert in Q(sqrt(5))), A045468 (primes decomposing), A003631 (primes remaining inert), A042993 (primes not decomposing).

Kronecker symbols {(D/n)} for negative fundamental discriminants D = -3..-47, -67, -163: A102283, A101455, A175629, A188510, A011582, A316569, A011585, A289741, A011586, A109017, A011588, A390614, A388073, A388072, A011591, A011592, A011596, A011615.

Kronecker symbols {(D/n)} for positive fundamental discriminants D = 5..41: this sequence, A091337, A110161, A011583, A011584, A322829, A322796, A390615, A011587, A391502, A011589, A391503, A011590.

Sequence in context: A332828 A100047 A226162 * A011558 A276398 A204549

Adjacent sequences: A080888 A080889 A080890 * A080892 A080893 A080894

Keyword

sign,mult,easy

Author

N. J. A. Sloane, Sep 23 2003

Extensions

Name specified by Wolfdieter Lang, Feb 26 2014

Status

approved