A099013 a(n) = Sum_{k=0..n} 3^(k-1)*Fibonacci(k).

0, 1, 4, 22, 103, 508, 2452, 11929, 57856, 280930, 1363495, 6618856, 32128024, 155953777, 757013548, 3674624638, 17836995847, 86582609284, 420280790476, 2040085854985, 9902784679240, 48069126732586, 233332442310919

Offset: 0

Author

Paul Barry, Sep 22 2004

Keywords

• nonn
• easy

Comments

Partial sums of A099012. Binomial transform of A063092 (with leading 0).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,6,-9).

Crossrefs

Cf. A000045, A014335.

Programs

Magma

I:=[0, 1, 4]; [n le 3 select I[n] else 4*Self(n-1)+6*Self(n-2)-9*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 25 2012

Mathematica

Join[{a=0,b=1},Table[c=3*b+9*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
Table[Sum[3^(k-1) Fibonacci[k],{k,0,n}],{n,0,30}] (* or *) LinearRecurrence[{4,6,-9},{0,1,4},30] (* Harvey P. Dale, Dec 09 2011 *)
CoefficientList[Series[x/((1-x)(1-3x-9x^2)),{x,0,40}],x] (* Vincenzo Librandi, Jun 25 2012 *)

PARI

x='x+O('x^30); concat([0], Vec(x/((1-x)*(1 - 3*x - 9*x^2)))) \\ G. C. Greubel, Dec 31 2017

Formula

G.f.: x/((1-x)*(1 - 3*x - 9*x^2)).

a(n) = 4*a(n-1) + 6*a(n-2) - 9*a(n-3).

a(n) = 3^(n-1)*Sum_{k=0..n} Fibonacci(n-k)*3^(-k).

a(n) = (3/2 + 3*sqrt(5)/2)^n*(1/22 + 7*sqrt(5)/110) + (1/22 - 7*sqrt(5)/110)*(3/2 - 3*sqrt(5)/2)^n - 1/11.

a(n) = (3^n*A000285(n) - 1)/11, the case m = 3 of Sum_{k=0..n} m^(k-1)*F(k) = (m^n*(m*F(n) + F(n+1)) - 1)/(m^2 + m - 1), F=A000045. - Ehren Metcalfe, Apr 29 2018

Extensions

Sign in second formula corrected by Harvey P. Dale, Dec 09 2011