cp's OEIS Frontend

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

%I A011558 #75 Dec 14 2023 06:02:32
%S A011558 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,
%T A011558 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,
%U A011558 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
%N A011558 Expansion of (x + x^3)/(1 + x + ... + x^4) mod 2.
%C A011558 Multiplicative with a(5^e) = 0, a(p^e) = 1 otherwise. - _David W. Wilson_, Jun 12 2005
%C A011558 Characteristic function of numbers coprime to 5. - _Reinhard Zumkeller_, Nov 30 2009
%C A011558 From _R. J. Mathar_, Jul 15 2010: (Start)
%C A011558 The sequence is the principal Dirichlet character mod 5. (The other real character mod 5 is A080891.)
%C A011558 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)
%C A011558 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
%C A011558 Conjecture: a(n+1) is the number of ways of partitioning n into distinct parts of A084215. - _R. J. Mathar_, Mar 01 2023
%D A011558 Arthur Gill, Linear Sequential Circuits, McGraw-Hill, 1966, Eq. (17-10).
%D A011558 K. Mahler, p-adic numbers and their functions, 2nd ed., Cambridge University press, 1981.
%H A011558 Antti Karttunen, <a href="/A011558/b011558.txt">Table of n, a(n) for n = 0..65537</a>
%H A011558 Michael Gilleland, <a href="/selfsimilar.html">Some Self-Similar Integer Sequences</a>
%H A011558 R. Gold, <a href="http://dx.doi.org/10.1137/0114079">Characteristic linear sequences and their coset functions</a>, J. SIAM Applied. Math., 14 (1966), 980-985.
%H A011558 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%H A011558 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1).
%F A011558 O.g.f.: x*(1+x+x^2+x^3)/(1-x^5). - _Wolfdieter Lang_, Feb 05 2009
%F A011558 From _Reinhard Zumkeller_, Nov 30 2009: (Start)
%F A011558 a(n) = 1 - A079998(n).
%F A011558 a(A047201(n))=1, a(A008587(n))=0.
%F A011558 A033437(n) = Sum_{k=0..n} a(k)*(n-k). (End)
%F A011558 a(n) = n^4 mod 5. - _Gary Detlefs_, Mar 20 2010
%F A011558 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
%F A011558 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
%F A011558 a(n) = sgn(n mod 5). - _Wesley Ivan Hurt_, Jun 30 2013
%F A011558 Euler transform of length 5 sequence [ 1, 0, 0, -1, 1]. - _Michael Somos_, May 24 2015
%F A011558 Moebius transform is length 5 sequence [ 1, 0, 0, 0, -1]. - _Michael Somos_, May 24 2015
%F A011558 G.f.: f(x) - f(x^5) where f(x) := x / (1 - x). - _Michael Somos_, May 24 2015
%F A011558 |a(n)| = |A080891(n)| = |A100047(n)|. - _Michael Somos_, May 24 2015
%e A011558 G.f. = x + x^2 + x^3 + x^4 + x^6 + x^7 + x^8 + x^9 + x^11 + x^12 + ...
%p A011558 seq(n&^4 mod 5, n=0..50); # _Gary Detlefs_, Mar 20 2010
%t A011558 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 *)
%t A011558 a[ n_] := Sign@Mod[ n, 5]; (* _Michael Somos_, May 24 2015 *)
%o A011558 (PARI) a(n)=!!(n%5) \\ _Charles R Greathouse IV_, Sep 23 2012
%o A011558 (PARI) {a(n) = n%5>0}; /* _Michael Somos_, May 24 2015 */
%o A011558 (Scheme) (define (A011558 n) (if (zero? (modulo n 5)) 0 1)) ;; _Antti Karttunen_, Dec 21 2017
%Y A011558 Cf. A000035, A011655, A109720 coprimality with 2, 3, 7, respectively.
%Y A011558 Cf. A168185, A145568, A168184, A168182, A168181, A097325, A166486.
%Y A011558 Cf. A080891, A100047.
%K A011558 nonn,mult,easy
%O A011558 0,1
%A A011558 _N. J. A. Sloane_
%E A011558 More terms from _Antti Karttunen_, Dec 21 2017