A257170 Expansion of (1 + x) * (1 + x^3) / (1 + x^4) in powers of x.

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

Offset: 0

Author

Michael Somos, Apr 17 2015

Keywords

• sign
• mult
• easy

Examples

			G.f. = 1 + x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 + ...
		

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, -1).

Crossrefs

Cf. A188510.

Programs

Magma

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+x^3)/(1+x^4))); // G. C. Greubel, Aug 02 2018

Mathematica

a[ n_] := If[ EvenQ[ n], Boole[n == 0], (-1)^Quotient[ n, 4]];
a[ n_] := If[ n == 0, 1, Sign[ n] SeriesCoefficient[ (1 + x) * (1 + x^3) / (1 + x^4), {x, 0, Abs @ n}]];
CoefficientList[Series[(1+x)*(1+x^3)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)

PARI

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

PARI

{a(n) = if( n==0, 1, sign(n) * polcoeff( (1 + x) * (1 + x^3) / (1 + x^4), + x* O(x^abs(n)), abs(n)))};

PARI

x='x+O('x^60); Vec((1+x)*(1+x^3)/(1+x^4)) \\ G. C. Greubel, Aug 02 2018

Formula

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

a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e otherwise and a(0) = 1.

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

a(n) = A188510(n) unless n = 0.

a(n+1) - a(n) = (-1)^n if n>0.

G.f.: (1 + x) * (1 + x^3) / (1 + x^4) = 1 + (x + x^3) / (1 + x^4).

G.f.: (1 - x^2) * (1 - x^4) * (1 - x^6) / ((1 - x) * (1 - x^3) * (1 - x^8)).

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