A117576 Expansion of (1-x^3)/((1-x^2)*(1+2*x^2)).

1, 0, -1, -1, 3, 1, -5, -3, 11, 5, -21, -11, 43, 21, -85, -43, 171, 85, -341, -171, 683, 341, -1365, -683, 2731, 1365, -5461, -2731, 10923, 5461, -21845, -10923, 43691, 21845, -87381, -43691, 174763, 87381, -349525, -174763, 699051, 349525, -1398101, -699051

Offset

0,5

Comments

A signed pair-reversal of the Jacobsthal numbers A001045.

Diagonal sums of A116949.

Links

Table of n, a(n) for n=0..43.

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

Formula

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

a(n) = -a(n-1) - 2*a(n-2) - 2*a(n-3).

a(n) = 2^(n/2)*(2*cos(Pi*n/2)/3 + sqrt(2)*sin(Pi*n/2)/6) + (-1)^n/3.

a(n) = floor(((-1)^(floor(n/2))*2^(2*floor(n/2)+1-floor((n+1)/2))+1)/3). - Tani Akinari, Nov 09 2012

E.g.f.: (4*cos(sqrt(2)*x) + 2*cosh(x) + sqrt(2)*sin(sqrt(2)*x) - 2*sinh(x)) / 6. - Amiram Eldar, Jan 15 2026

Mathematica

LinearRecurrence[{-1, -2, -2}, {1, 0, -1}, 50] (* Amiram Eldar, Jan 15 2026 *)

Crossrefs

Cf. A001045, A112447, A116949.

Sequence in context: A077881 A129095 A105604 * A112447 A289360 A290212

Adjacent sequences: A117573 A117574 A117575 * A117577 A117578 A117579

Keyword

easy,sign

Author

Paul Barry, Mar 29 2006

Status

approved