A226328 a(0)=1, a(1)=-2; a(n+2) = a(n+1) + a(n) + (period 3: repeat 3, 1, -1).

1, -2, 2, 1, 2, 6, 9, 14, 26, 41, 66, 110, 177, 286, 466, 753, 1218, 1974, 3193, 5166, 8362, 13529, 21890, 35422, 57313, 92734, 150050, 242785, 392834, 635622, 1028457, 1664078, 2692538, 4356617, 7049154, 11405774, 18454929, 29860702, 48315634, 78176337

Offset

0,2

Comments

a(n+1)/a(n) -> the golden ratio, A001622.

a(3*n)+a(3*n+1)+a(3*n+2) = 1,9,49,217,929,... = b(n), and b(n+1)-b(n) = 8*A015448(n+1).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

a(n) = F(n-3) + F(n) - A010872(n+1).

a(n+3) = a(n) + 4*F(n).

G.f.: (2*x^4+3*x^2-3*x+1)/( (x-1)*(x^2+x-1)*(1+x+x^2) ). [Charles R Greathouse IV, Jun 04 2013]

a(n) = A057078(n+1) +2*A212804(n) -1. - R. J. Mathar, Jun 26 2013

Example

a(2)=-2+1+3=2, a(3)=2-2+1=1, a(4)=1+2-1=2, a(5)=2+1+3=6.
a(0)=F(-3)+F(n)-1=2+0-1=1,  a(1)=-1+1-2=-2, a(2)=1+1-0=2.
a(3)=1+4*0=1, a(4)=-2+4*1=2, a(5)=2+4*1=6, a(6)=1+4*2=9.

Mathematica

CoefficientList[Series[(2 x^4 + 3 x^2 - 3 x + 1) / (x^5 + x^4 - x^3 - x^2 - x + 1), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 05 2013 *)
LinearRecurrence[{1, 1, 1, -1, -1}, {1, -2, 2, 1, 2}, 40] (* Hugo Pfoertner, Feb 12 2024 *)

Prog

(PARI) Vec((2*x^4+3*x^2-3*x+1)/(x^5+x^4-x^3-x^2-x+1)+O(x^99)) \\ Charles R Greathouse IV, Jun 04 2013
(Magma) I:=[1, -2, 2, 1, 2]; [n le 5 select I[n] else Self(n-1)+Self(n-2)+Self(n-3)-Self(n-4)-Self(n-5): n in [1..40]]; // Vincenzo Librandi, Jun 05 2013

Crossrefs

Cf. A156279, A022120, A022353.

Sequence in context: A368836 A336823 A236144 * A307599 A162663 A005007

Adjacent sequences: A226325 A226326 A226327 * A226329 A226330 A226331

Keyword

sign,easy,less

Author

Paul Curtz, Jun 04 2013

Extensions

a(23) corrected by Charles R Greathouse IV, Jun 04 2013

More terms from Bruno Berselli, Jun 04 2013

Status

approved