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A257170 Expansion of (1 + x) * (1 + x^3) / (1 + x^4) in powers of x. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

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)))))))).

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 *)

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) 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: A322860 A178225 A264739 * A073097 A117569 A135528

Adjacent sequences:  A257167 A257168 A257169 * A257171 A257172 A257173

KEYWORD

sign,mult,easy

AUTHOR

Michael Somos, Apr 17 2015

STATUS

approved

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Last modified September 29 21:59 EDT 2020. Contains 337432 sequences. (Running on oeis4.)