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 A112299 Expansion of x * (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^8) in powers of x. 3
 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Periodic with period length 8. Sum_{k>=1} a(k)/k = Pi/8. - Jaume Oliver Lafont, Oct 20 2009 LINKS G. C. Greubel, Table of n, a(n) for n = 1..2500 Index entries for linear recurrences with constant coefficients, signature (0,-1,0,-1,0,-1). FORMULA Euler transform of length 8 sequence [-1, -1, -1, 0, 0, 0, 0, 1]. 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). G.f.: x * (1 + x + x^2) * (1 - x)^2 / ((1 + x^2) * (1 + x^4)). G.f.: f(x) - f(x^2) where f(x) := x / (1 + x^2). - Michael Somos, Jun 19 2015 a(n) = -a(4 - n) = a(n + 8) for all n in Z. a(4*n) = 0. a(2*n) = - A056594(n-1). a(2*n + 1) = A033999(n). a(4*n + 1) = 1. a(4*n + 3) = -1. a(4*n + 2) = - A033999(n). - Michael Somos, Jun 19 2015 a(n) = A257196(n) unless n=0. - Michael Somos, Sep 01 2015 EXAMPLE G.f. = x - x^2 - x^3 + x^5 + x^6 - x^7 + x^9 - x^10 - x^11 + x^13 + x^14 - x^15 + ... MATHEMATICA LinearRecurrence[{0, -1, 0, -1, 0, -1}, {1, -1, -1, 0, 1, 1}, 110] (* Harvey P. Dale, Dec 07 2014 *) a[ n_] := {1, -1, -1, 0, 1, 1, -1, 0}[[Mod[n, 8, 1]]]; PROG (PARI) {a(n) = [0, 1, -1, -1, 0, 1, 1, -1][n%8 + 1]}; (PARI) {a(n) = [0, 1, -(-1)^(n\4), -1][n%4 + 1]}; /* Michael Somos, Jun 19 2015 */ (MAGMA) m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x+x^2)*(1-x)^2/((1+x^2)*(1+x^4)))); // G. C. Greubel, Aug 03 2018 CROSSREFS Cf. A033999, A056594, A257196. Sequence in context: A076213 A120525 A285373 * A230901 A285671 A267773 Adjacent sequences:  A112296 A112297 A112298 * A112300 A112301 A112302 KEYWORD sign,mult,easy AUTHOR Michael Somos, Sep 02 2005 STATUS approved

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Last modified January 19 12:38 EST 2021. Contains 340269 sequences. (Running on oeis4.)