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A186111 a(n) = -n if n odd, a(2n) = 3n if n odd, a(4n) = 2n. 3
1, -3, 3, -2, 5, -9, 7, -4, 9, -15, 11, -6, 13, -21, 15, -8, 17, -27, 19, -10, 21, -33, 23, -12, 25, -39, 27, -14, 29, -45, 31, -16, 33, -51, 35, -18, 37, -57, 39, -20, 41, -63, 43, -22, 45, -69, 47, -24, 49, -75, 51, -26, 53, -81, 55, -28, 57, -87, 59, -30, 61, -93, 63 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..10000

M. Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (-2,-3,-4,-3,-2,-1)

FORMULA

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

Euler transform of length 4 sequence [-3, 0, -1, 2].

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

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

Dirichlet g.f. zeta(s-1)*( 1-5*2^(-s)+4^(1-s)). - R. J. Mathar, Mar 31 2011

a(n) = (-1)^(n+1)*n + (-1)^floor(n/2)*A027656(n-2). - R. J. Mathar, Mar 31 2011

a(n) = -2*a(n-1) - 3*a(n-2) - 4*a(n-3) - 3*a(n-4) - 2*a(n-5) - a(n-6) with a(1)=1, a(2)=-3, a(3)=3, a(4)=-2, a(5)=5, a(6)=-9. - Harvey P. Dale, Aug 08 2012

G.f.: 1/(1+x) - 1/(1+x)^2 - 1/(1+x^2) + 1/(1+x^2)^2. - Michael Somos, Apr 24 2015

a(n) = -a(-n) for all n in Z. - Michael Somos, Apr 24 2015

G.f.: f(x) - f(x^2) where f(x) := x / (1 + x)^2. - Michael Somos, May 07 2015

Moebius transform of A186690. - Michael Somos, Apr 25 2015

a(n) = -(-1)^n * A186813(n). - Michael Somos, May 07 2015

EXAMPLE

G.f. = x - 3*x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 9*x^6 + 7*x^7 - 4*x^8 + 9*x^9 + ...

MATHEMATICA

Rest[CoefficientList[Series[x (1-x)^3(1-x^3)/(1-x^4)^2, {x, 0, 70}], x]] (* or *) LinearRecurrence[{-2, -3, -4, -3, -2, -1}, {1, -3, 3, -2, 5, -9}, 70] (* Harvey P. Dale, Aug 08 2012 *)

a[ n_] := n If[ OddQ[n], 1, -(Mod[n/2, 2] + 1/2)]; (* Michael Somos, Apr 25 2015 *)

a[ n_] := n {1, -3/2, 1, -1/2}[[Mod[n, 4, 1]]]; (* Michael Somos, Apr 25 2015 *)

PROG

(PARI) {a(n) = -(-1)^n * n * [1, 2, 3, 2] [n%4 + 1] / 2};

(PARI) {a(n) = sign(n) * polcoeff( x * (1 - x)^3 * (1 - x^3) / (1 - x^4)^2 + x * O(x^abs(n)), abs(n))};

(PARI) {a(n) = n * if( n%2, 1, -(n/2%2 + 1)/2)}; /* Michael Somos, Apr 25 2015 */

(MAGMA) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1-x)^3*(1-x^3)/(1-x^4)^2)); // G. C. Greubel, Aug 14 2018

CROSSREFS

Cf. A068073, A186690, A186813.

Sequence in context: A070163 A083343 A292527 * A186813 A293521 A285443

Adjacent sequences:  A186108 A186109 A186110 * A186112 A186113 A186114

KEYWORD

sign,mult,easy

AUTHOR

Michael Somos, Feb 13 2011

STATUS

approved

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Last modified October 22 07:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)