A257196 Expansion of (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)) in powers of x.

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

Offset: 0

Author

Michael Somos, Apr 17 2015

Keywords

• sign
• mult
• easy

Examples

			G.f. = 1 + x - x^2 - x^3 + x^5 + x^6 - x^7 + x^9 - x^10 - x^11 + x^13 + ...
		

Links

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

Crossrefs

Cf. A112299.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4)))); // G. C. Greubel, Aug 02 2018

Mathematica

a[ n_] := Boole[n == 0] + {1, -1, -1, 0, 1, 1, -1, 0}[[Mod[ n, 8, 1]]];
a[ n_] := If[ n == 0, 1, Sign[ n] SeriesCoefficient[ (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)), {x, 0, Abs @ n}]];
CoefficientList[Series[(1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4)), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
LinearRecurrence[{0,-1,0,-1,0,-1},{1,1,-1,-1,0,1,1},100] (* Harvey P. Dale, Nov 16 2022 *)

PARI

{a(n) = (n==0) + [0, 1, -1, -1, 0, 1, 1, -1][n%8 + 1]};

PARI

{a(n) = if( n==0, 1, n%2, (-1)^(n\2), n%4 == 2, -(-1)^(n\4), 0)};

PARI

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)) + x * O(x^abs(n)), abs(n)))};

PARI

x='x+O('x^60); Vec((1 + x)*(1 + x^5)/((1 + x^2)*(1 + x^4))) \\ G. C. Greubel, Aug 02 2018

Formula

Euler transform of length 10 sequence [1, -2, 0, 0, 1, 0, 0, 1, 0, -1].

a(n) is multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e) = 1 if p == 1 (mod 4), a(p^e) = (-1)^e if p == 3 (mod 4) and a(0) = 1.

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

G.f.: (1 + x) * (1 + x^5) / ((1 + x^2) * (1 + x^4)).

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

a(n) = A112299(n) unless n=0. - R. J. Mathar, Apr 19 2015