A257179 Expansion of (1 + x^5) / ((1 - x) * (1 + x^4)) in powers of x.

1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 1, 0, 1, 1, 1, 2, 1

Offset

0,9

Links

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

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

Formula

Euler transform of length 10 sequence [1, 0, 0, -1, 1, 0, 0, 1, 0, -1].

Moebius transform is length 8 sequence [1, 0, 0, -1, 0, 0, 0, 2].

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

G.f.: (1 + x^5) / ((1 - x) * (1 + x^4)).

G.f.: (1 - x^4) * (1 - x^10) / ((1 - x) * (1 - x^5) * (1 - x^8)).

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

a(n) = a(-n) for all n in Z. a(n+8) = a(n) unless n=0 or n=-8. a(8*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1. a(8*n + 4) = 0.

a(n) = A259042(n+4) unless n = 0.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. - Amiram Eldar, Nov 20 2022

Dirichlet g.f.: zeta(s)*(1-1/4^s+2/8^s). - Amiram Eldar, Jan 05 2023

Example

G.f. = 1 + x + x^2 + x^3 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + x^11 + ...

Mathematica

a[ n_] := Boole[n != 0] - Boole[Mod[n, 4] == 0] + 2 Boole[Mod[n, 8] == 0];
a[ n_] := -Boole[n == 0] + {1, 1, 1, 0, 1, 1, 1, 2}[[Mod[n, 8, 1]]];
a[ n_] := SeriesCoefficient[ (1 + x^5) / ((1 - x) * (1 + x^4)), {x, 0, Abs@n}];

Prog

(PARI) {a(n) = (n != 0) - (n%4 == 0) + 2*(n%8 == 0)};
(PARI) {a(n) = -(n==0) + [2, 1, 1, 1, 0, 1, 1, 1][n%8 + 1]};
(PARI) {a(n) = polcoeff( (1 + x^5) / ((1 - x) * (1 + x^4)) + x * O(x^abs(n)), abs(n))};

Crossrefs

Cf. A010877, A259042.

Sequence in context: A326695 A343749 A323191 * A259042 A350074 A333179

Adjacent sequences: A257176 A257177 A257178 * A257180 A257181 A257182

Keyword

nonn,mult,easy

Author

Michael Somos, Apr 17 2015

Extensions

More terms from Antti Karttunen, Jul 29 2018

Status

approved