login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A011558 Expansion of (x+x^3)/(1+x+...+x^4) mod 2. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - David W. Wilson, Jun 12 2005

From Reinhard Zumkeller, Nov 30 2009: (Start)

a(n)=1-A079998(n); characteristic function of numbers coprime to 5; a(A047201(n))=1; a(A008587(n))=0;

A033437(n) = SUM(a(k)*(n-k): 0<=k<=n). (End)

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

The sequence is the principal Dirichlet character mod 5 (The other real character mod 5 is A080891.)

Associated Dirichlet L-functions are for example L(2,chi)= sum_{n>=1} a(n)/n^2 = 1.5791367... = (psi'(1/5)+psi'(2/5)+psi'(3/5)+psi'(4/5))/25 or L(3,chi)= sum_{n>=1} a(n)/n^3 = 1.192440... = -(psi''(1/5)+psi''(2/5)+psi''(3/5)+psi''(4/5))/250, where psi' and psi'' are the trigamma and tetragamma functions. (End)

a(n) is for n >= 1 also the characteristic function for rational g-adic integers (+n/5)_g and also (-n/5)_g for all integers g >= 2 without a factor of 5 (A047201). See the definition in the Mahler reference, p. 7 and also p. 10. - Wolfdieter Lang, Jul 11 2014

REFERENCES

Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).

K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.

LINKS

Table of n, a(n) for n=0..80.

Michael Gilleland, Some Self-Similar Integer Sequences

R. Gold, Characteristic linear sequences and their coset functions, J. SIAM Applied. Math., 14 (1966), 980-985.

Index entries for characteristic functions

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

FORMULA

O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - Wolfdieter Lang, Feb 05 2009

a(n)=(1/8)*{3*(n mod 4)+[(n+1) mod 4]+[(n+2) mod 4]-[(n+3) mod 4]}, with n>=0. - Paolo P. Lava, Feb 11 2009

a(n)=n^4 mod 5. - From Gary Detlefs, Mar 20 2010

Sum_{n=1..infinity} a(n)/n^s = L(s,chi) = (1-1/5^s)*Riemann_zeta(s), s>1. - R. J. Mathar, Jul 31 2010

For the general case. The characteristic function of numbers that are not multiples of m is a(n)=floor((n-1)/m)-floor(n/m)+1, m,n > 0. - Boris Putievskiy, May 08 2013

a(n) = sgn(n mod 5). - Wesley Ivan Hurt, Jun 30 2013

Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - Michael Somos, May 24 2015

Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - Michael Somos, May 24 2015

G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - Michael Somos, May 24 2015

|a(n)| = |A080891(n)| = |A100047(n)|. - Michael Somos, May 24 2015

EXAMPLE

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

MAPLE

seq(n^4 mod 5, n=0..50); # [From Gary Detlefs, Mar 20 2010]

MATHEMATICA

Mod[#, 2]&/@CoefficientList[Series[(x+x^3)/(1+x+x^2+x^3+x^4) , {x, 0, 100}], x] (* or *) Flatten[Table[{0, 1, 1, 1, 1}, {30}]] (* Harvey P. Dale, May 15 2011 *)

a[ n_] := Sign@Mod[ n, 5]; (* Michael Somos, May 24 2015 *)

PROG

(PARI) a(n)=!!(n%5) \\ Charles R Greathouse IV, Sep 23 2012

(PARI) {a(n) = n%5>0}; /* Michael Somos, May 24 2015 */

CROSSREFS

Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively.

Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486.

Cf. A080891, A100047.

Sequence in context: A092248 A106743 A244895 * A100047 A080891 A112713

Adjacent sequences:  A011555 A011556 A011557 * A011559 A011560 A011561

KEYWORD

nonn,mult,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified July 22 18:10 EDT 2017. Contains 289671 sequences.