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

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.

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

Example

G.f. = x - x^2 - x^3 + x^4 + x^6 - x^7 - x^8 + x^9 + x^11 - x^12 - x^13 + ...

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.

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