A154691 Expansion of (1+x+x^2) / ((1-x-x^2)*(1-x)).

1, 3, 7, 13, 23, 39, 65, 107, 175, 285, 463, 751, 1217, 1971, 3191, 5165, 8359, 13527, 21889, 35419, 57311, 92733, 150047, 242783, 392833, 635619, 1028455, 1664077, 2692535, 4356615, 7049153, 11405771, 18454927, 29860701, 48315631, 78176335

Offset

0,2

Links

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

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Formula

a(n+1) - a(n) = A006355(n+3) = A055389(n+3).

a(n) = A066629(n-1) + A066629(n).

a(n) = A006355(n+4) - 3 = A078642(n+1) - 3.

a(n+1) = a(n) + 2*A000045(n+2). - Reinhard Zumkeller, Nov 17 2013

From Colin Barker, Feb 01 2017: (Start)

a(n) = -3 + (2^(1-n)*((1-r)^n*(-2+r) + (1+r)^n*(2+r))) / r where r=sqrt(5).

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

(End)

a(n) = 2*Fibonacci(n+3) - 3. - Greg Dresden, Oct 10 2020

Maple

A154691 := proc(n) coeftayl( (1+x+x^2)/(1-x-x^2)/(1-x), x=0, n) ; end proc:

Mathematica

Fibonacci[Range[3, 5! ]]*2-3 (* Vladimir Joseph Stephan Orlovsky, Mar 19 2010 *)
CoefficientList[Series[(1 + x + x^2)/((1 - x - x^2)(1 - x)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 18 2012 *)

Prog

(Haskell)
a154691 n = a154691_list !! n
a154691_list = 1 : zipWith (+)
                   a154691_list (drop 2 $ map (* 2) a000045_list)
-- Reinhard Zumkeller, Nov 17 2013
(PARI) Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ Colin Barker, Feb 01 2017

Crossrefs

Cf. A000045, A006355, A055389, A066629, A078642.

Cf. A001595, A166863.

Sequence in context: A136851 A155339 A122886 * A306902 A227121 A078447

Adjacent sequences: A154688 A154689 A154690 * A154692 A154693 A154694

Keyword

easy,nonn

Author

R. J. Mathar, Jan 14 2009

Status

approved