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A099837 Expansion of (1 - x^2) / (1 + x + x^2) in powers of x. 32
1, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1, 2, -1, -1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A transform of (-1)^n.

Row sums of Riordan array ((1-x)/(1+x),x/(1+x)^2), A110162.

Let b(n)=sum{k=0..floor(n/2), binomial(n-k,k)(-1)^(n-2k)}. Then a(n)=b(n)-b(n-2)=A049347(n)-A049347(n-2) (n>0). The g.f. 1/(1+x) of (-1)^n is transformed to (1-x^2)/(1+x+x^2) under the mapping G(x)->((1-x^2)/(1+x^2))G(x/(1+x^2)). Partial sums of A099838.

A(n) = a(n+3) (or a(n) if a(0) is replaced by 2) appears, together with B(n) = A049347(n) in the formula 2*exp(2*Pi*n*I/3) = A(n) + B(n)*sqrt(3)*I, n >= 0, with I = sqrt(-1).  See A164116 for the case N=5. - Wolfdieter Lang, Feb 27 2014

LINKS

Table of n, a(n) for n=0..71.

M. Somos, Rational Function Multiplicative Coefficients

Index entries for linear recurrences with constant coefficients, signature (-1,-1).

FORMULA

G.f.: (1-x^2)/(1+x+x^2); a(n)=sum{j=0..n, (C(1, j/2)(-1)^(j/2)(1+(-1)^j)/2)*sum{k=0..floor((n-j)/2), C(n-j-k, k)(-1)^(n-j-k)}}; a(n)=2cos(2*Pi*n/3)-0^n.

Euler transform of length 3 sequence [ -1, -1, 1]. - Michael Somos, Mar 21 2011

Moebius transform is length 3 sequence [ -1, 0, 3]. - Michael Somos, Mar 22 2011

a(n) = -b(n) where b(n) = A061347(n) is multiplicative with b(3^e) = -2 if e>0, b(p^e) = 1 otherwise. - Michael Somos, Jan 19 2012

a(n) = a(-n). a(n) = c_3(n) if n>1 where c_k(n) is Ramanujan's sum. - Michael Somos, Mar 21 2011

G.f.: (1 - x) * (1 - x^2) / (1 - x^3). a(n) = -a(n-1) - a(n-2) unless n=0, 1, 2. - Michael Somos, Jan 19 2012

Dirichlet g.f. sum_{n>=1} a(n)/n^s = zeta(s)*(3^(1-s)-1). - R. J. Mathar, Apr 11 2011

a(n+3) = R(n,-1) for n>=0, with the monic Chebyshev T-polynomials R with coefficient table A127672. - Wolfdieter Lang, Feb 27 2014

EXAMPLE

G.f. = 1 - x - x^2 + 2*x^3 - x^4 - x^5 + 2*x^6 - x^7 - x^8 + 2*x^9 - x^10 + ...

MATHEMATICA

a[0] = 1; a[n_] := Mod[n+2, 3] - Mod[n, 3]; A099837 = Table[a[n], {n, 0, 71}](* Jean-Fran├žois Alcover, Feb 15 2012, after Michael Somos *)

PROG

(PARI) {a(n) = [2, -1, -1][n%3 + 1] - (n == 0)}; /* Michael Somos, Jan 19 2012 */

(Maxima) A099837(n) := block(

        if n = 0 then 1 else [2, -1, -1][1+mod(n, 3)]

)$ /* R. J. Mathar, Mar 19 2012 */

CROSSREFS

Cf. A061347, A100051, A100063, A098554.

Sequence in context: A057559 A205375 A016010 * A100051 A122876 A131713

Adjacent sequences:  A099834 A099835 A099836 * A099838 A099839 A099840

KEYWORD

easy,sign

AUTHOR

Paul Barry, Oct 27 2004

STATUS

approved

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Last modified December 7 11:34 EST 2016. Contains 278874 sequences.