A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.

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

Author

N. J. A. Sloane

Keywords

• nonn
• mult
• easy

Comments

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

Characteristic function of numbers coprime to 5. - Reinhard Zumkeller, Nov 30 2009

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)A047201).%20See%20the%20definition%20in%20the%20Mahler%20reference,%20p.%207%20and%20also%20p.%2010.%20-%20_Wolfdieter%20Lang">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

Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - R. J. Mathar, Mar 01 2023

Examples

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

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

• Antti Karttunen, Table of n, a(n) for n = 0..65537
• 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).

Crossrefs

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

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

Cf. A080891, A100047.

Programs

Maple

seq(n&^4 mod 5, n=0..50); # 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 *)

PARI

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

PARI

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

Scheme

(define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; Antti Karttunen, Dec 21 2017

Formula

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

From Reinhard Zumkeller, Nov 30 2009: (Start)

a(n) = 1 - A079998(n).

a(A047201(n))=1, a(A008587(n))=0.

A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End)

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

Sum_{n>=1} 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

Extensions

More terms from Antti Karttunen, Dec 21 2017