A259044 Period 8 sequence [ 0, 1, 0, 1, 1, 1, 0, 1, ...].

0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0

Offset

0,1

Links

Table of n, a(n) for n=0..86.

Michael Somos, Rational Function Multiplicative Coefficient.

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

Formula

Euler transform of length 8 sequence [0, 1, 1, 0, -1, -1, 0, 1].

Moebius transform is length 8 sequence [1, -1, 0, 1, 0, 0, 0, -1].

a(n) is multiplicative with a(2) = 0, a(4) = 1, a(2^e) = 0 if e>2, a(p^e) = 1 if p>2.

G.f.: x * (1 - x^5) * (1 + x^3) / ((1 - x^2) * (1 - x^8)).

G.f.: x * (1 - x^5) * (1 - x^6) / ((1 - x^2) * (1 - x^3) * (1 - x^8)).

G.f.: f(x) + f(x^4) where f(x) := x / (1 - x^2).

a(n) = a(-n) = a(n+8) for all n in Z.

a(n) = -(-1)^n * A280237(n). a(2*n + 1) = 1. a(4*n + 2) = 0.

G.f.: -x*(x^2-x+1)*(x^4+x^3+x^2+x+1) / ( (x-1)*(1+x)*(x^2+1)*(x^4+1) ). - R. J. Mathar, Jun 18 2015

Dirichlet g.f.: zeta(s)*(1-1/2^s+1/4^s-1/8^s). - Amiram Eldar, Jan 01 2023

Example

G.f. = x + x^3 + x^4 + x^5 + x^7 + x^9 + x^11 + x^12 + x^13 + x^15 + ...

Mathematica

a[ n_] := Mod[n, 2] + Boole[Mod[n, 8] == 4];
a[ n_] := {1, 0, 1, 1, 1, 0, 1, 0}[[Mod[n, 8, 1]]];
a[ n_] := SeriesCoefficient[ x / (1 - x^2) + x^4 / (1 - x^8), {x, 0, Abs@n}];
PadRight[{}, 120, {0, 1, 0, 1, 1, 1, 0, 1}] (* Harvey P. Dale, Feb 22 2018 *)

Prog

(PARI) {a(n) = (n%2) + (n%8==4)};
(PARI) {a(n) = [0, 1, 0, 1, 1, 1, 0, 1][n%8 + 1]};
(PARI) {a(n) = polcoeff( x / (1 - x^2) + x^4 / (1 - x^8) + x * O(x^abs(n)), abs(n))};

Crossrefs

Cf. A280237.

Sequence in context: A327180 A030308 A280237 * A285383 A276950 A285351

Adjacent sequences: A259041 A259042 A259043 * A259045 A259046 A259047

Keyword

nonn,mult,easy

Author

Michael Somos, Jun 17 2015

Status

approved