A257181 Expansion of (1 - x) * (1 + x^4) / (1 + x^5) in powers of x.

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

Offset: 0

Author

Michael Somos, Apr 17 2015

Keywords

• sign
• easy

Examples

			G.f. = 1 - x + x^4 - 2*x^5 + x^6 - x^9 + 2*x^10 - x^11 + x^14 - 2*x^15 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,-1).

Crossrefs

Cf. A164116, A257179.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x)*(1+x^4)/(1+x^5))); // G. C. Greubel, Aug 02 2018

Mathematica

a[ n_] := -Boole[n == 0] + {-1, 0, 0, 1, -2, 1, 0, 0, -1, 2}[[Mod[n, 10, 1]]];
a[ n_] := SeriesCoefficient[ (1 - x) * (1 + x^4) / (1 + x^5), {x, 0, Abs@n}];
CoefficientList[Series[(1-x)*(1+x^4)/(1+x^5), {x, 0, nmax}], x] (* G. C. Greubel, Aug 02 2018 *)

PARI

{a(n) = if( n==0, 1, (-1)^(n\5) * [2, -1, 0, 0, 1][n%5 + 1])};

PARI

{a(n) = polcoeff( (1 - x) * (1 + x^4) / (1 + x^5) + x * O(x^abs(n)), abs(n))};

PARI

x='x+O('x^60); Vec((1-x)*(1+x^4)/(1+x^5)) \\ G. C. Greubel, Aug 02 2018

Formula

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

a(n) = a(-n) for all n in Z. a(n+5) = -a(n) unless n = 0 or -5. a(5*n) = 2 * (-1)^n unless n = 0. a(5*n + 2) = a(5*n + 3) = 0. a(5*n + 1) = a(5*n - 1) = -(-1)^n.

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

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

Convolution inverse is A257179.

a(n) = (-1)^floor( (n+4) / 5) * A164116(n).