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

0,1

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).

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)))))))).

E.g.f.: 1 + sqrt(2) * cosh(x/sqrt(2)) * sin(x/sqrt(2)). - Amiram Eldar, Nov 27 2025

Example

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

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 *)
LinearRecurrence[{0, 0, 0, -1}, {1, 1, 0, 1, 0}, 80] (* Harvey P. Dale, Nov 20 2025 *)

Prog

(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) my(x='x+O('x^60)); Vec((1+x)*(1+x^3)/(1+x^4)) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x)*(1+x^3)/(1+x^4))); // G. C. Greubel, Aug 02 2018

Crossrefs

Cf. A188510.

Sequence in context: A373990 A264739 A373252 * A073097 A117569 A135528

Adjacent sequences: A257167 A257168 A257169 * A257171 A257172 A257173

Keyword

sign,mult,easy

Author

Michael Somos, Apr 17 2015

Status

approved