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

0,1

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)_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

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).

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

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

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: A100047 A226162 A080891 * A276398 A204549 A112713

Adjacent sequences: A011555 A011556 A011557 * A011559 A011560 A011561

Keyword

nonn,mult,easy

Author

N. J. A. Sloane

Extensions

More terms from Antti Karttunen, Dec 21 2017

Status

approved