A102517 Expansion of (1+x^2)/((1-x+x^2)(1+2x^2)).

1, 1, -1, -2, 1, 3, -2, -5, 5, 10, -11, -21, 22, 43, -43, -86, 85, 171, -170, -341, 341, 682, -683, -1365, 1366, 2731, -2731, -5462, 5461, 10923, -10922, -21845, 21845, 43690, -43691, -87381, 87382, 174763, -174763, -349526, 349525, 699051, -699050, -1398101, 1398101, 2796202, -2796203, -5592405

Offset

0,4

Links

Table of n, a(n) for n=0..47.

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

Formula

G.f.: (1+x^2)^2/((1+x^2)^3+x^6)+x(1+x^2)/((1+x^2)^3+x^6); a(n)=sum{k=0..floor(n/2), T(n-k, k)(-1)^k}, T(n, k)=sum{i=0..n, C(n, i)} (A008949); a(n)=(-1)^(n/2)sum{k=0..floor(n/6), C(n/2, 3k)}(1+(-1)^n)2+ (-1)^((n-1)/2)sum{k=0..floor((n+1)/6), C((n+1)/2, 3k+1)}(1-(-1)^n)/2; a(n)=2^(n/2)(cos(pi*n/2)/3+sqrt(2)sin(pi*n/2)/3)+cos(pi*n/3+pi/3)/3+sqrt(3)sin(pi*n/3+pi/3)/3; a(2n)=(-1)^n*A024493(n); a(2n+1)=(-1)^n*A024494(n).

a(0)=1, a(1)=1, a(2)=-1, a(3)=-2, a(n)=a(n-1)-3*a(n-2)+2*a(n-3)- 2*a(n-4) [From Harvey P. Dale, Oct 28 2011]

Mathematica

CoefficientList[Series[(1+x^2)/((1-x+x^2)(1+2x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{1, -3, 2, -2}, {1, 1, -1, -2}, 50] (* Harvey P. Dale, Oct 28 2011 *)

Crossrefs

Cf. A001045, A078008.

Sequence in context: A045931 A325193 A079974 * A062951 A263150 A145794

Adjacent sequences: A102514 A102515 A102516 * A102518 A102519 A102520

Keyword

easy,sign

Author

Paul Barry, Jan 13 2005

Status

approved