A138574 a(n) = 2*a(n-1) + 2*a(n-2) + 2*a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=3, a(3)=9, a(4)=25.

0, 1, 3, 9, 25, 73, 211, 609, 1761, 5089, 14707, 42505, 122841, 355017, 1026019, 2965249, 8569729, 24766977, 71577891, 206863945, 597847897, 1727812489, 4993470771, 14431398369, 41707515361, 120536956513, 348358269715, 1006774084809, 2909631106713, 8408989965961

Offset: 0

Author

Benoit Cloitre, May 12 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Crossrefs

Cf. A071101.

Programs

Magma

I:=[0,1,3,9,25]; [n le 5 select I[n] else 2*Self(n-1)+2*Self(n-2)+2*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 14 2018

Maple

seq(coeff(series((x*(1+x+x^2-x^3))/(1-2*x-2*x^2-2*x^3+x^4),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Sep 12 2018

Mathematica

LinearRecurrence[{2, 2, 2, -1}, {0, 1, 3, 9, 25}, 50] (* G. C. Greubel, Aug 08 2017 *)
CoefficientList[Series[x*( 1+x+x^2-x^3 )/(1-2*x-2*x^2-2*x^3+x^4), {x, 0, 20}], x] (* Stefano Spezia, Sep 12 2018 *)

PARI

x='x+O('x^50); concat([0], Vec(x*(1 +x +x^2 -x^3)/(1 -2*x -2*x^2 -2*x^3 +x^4))) \\ G. C. Greubel, Aug 08 2017

Formula

a(n) = round(w^n*(1 + 1/sqrt(5))/4) where w = (1+r)/(1-r) = 2.89005363826396... and r = sqrt(sqrt(5)-2) = 0.485868271756... .

G.f.: x*( 1 + x + x^2 - x^3 ) / ( 1 - 2*x - 2*x^2 - 2*x^3 + x^4 ). - R. J. Mathar, Jun 29 2013

Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018

Extensions

Terms corrected by Colin Barker, Jun 28 2013