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

Author

Michael Somos, Apr 17 2015

Keywords

• nonn
• mult
• easy

Examples

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

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).

Crossrefs

Cf. A010877, A259042.

Programs

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}];

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))};

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

Extensions

More terms from Antti Karttunen, Jul 29 2018