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

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

Author

R. J. Mathar, Jan 14 2009

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Tomislav Došlić and Biserka Kolarec, On Log-Definite Tempered Combinatorial Sequences, Mathematics (2025) Vol. 13, Iss. 7, 1179.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045, A001595, A006355, A055389, A066629, A078642, A166863.

Programs

Haskell

a154691 n = a154691_list !! n
a154691_list = 1 : zipWith (+)
                   a154691_list (drop 2 $ map (* 2) a000045_list)
-- Reinhard Zumkeller, Nov 17 2013

Magma

A154691:= func< n | 2*Fibonacci(n+3) - 3 >;
[A154691(n): n in [0..40]]; // G. C. Greubel, Jan 18 2025

Maple

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

Mathematica

Fibonacci[Range[3,60]]*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 *)

PARI

Vec((1+x+x^2) / ((1-x-x^2)*(1-x)) + O(x^60)) \\ Colin Barker, Feb 01 2017

Python

def A154691(n): return 2*fibonacci(n+3) - 3
print([A154691(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025

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

E.g.f.: 4*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 2*sqrt(5)*sinh(sqrt(5)*x/2))/5 - 3*exp(x). - Stefano Spezia, Apr 09 2025