A022308 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=0, a(1)=3.

0, 3, 4, 8, 13, 22, 36, 59, 96, 156, 253, 410, 664, 1075, 1740, 2816, 4557, 7374, 11932, 19307, 31240, 50548, 81789, 132338, 214128, 346467, 560596, 907064, 1467661, 2374726, 3842388, 6217115, 10059504, 16276620, 26336125, 42612746, 68948872, 111561619

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A000032, A000045, A000285, A122195.

Programs

Maple

with(combinat): seq(fibonacci(n)+fibonacci(n+5)-1, n=-2..30); # Zerinvary Lajos, Feb 01 2008

Mathematica

Table[(3 LucasL[n] - Fibonacci[n] - 2)/2, {n, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
LinearRecurrence[{2, 0, -1}, {0, 3, 4}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
CoefficientList[Series[x (3 - 2 x)/(x^3 - 2 x + 1), {x, 0, 20}], x] (* Eric W. Weisstein, Mar 26 2018 *)

PARI

concat(0, Vec(x*(3-2*x)/(x^3-2*x+1) + O(x^50))) \\ Colin Barker, Feb 20 2017

PARI

a(n) = if(n==0, 0, if(n==1, 3, a(n-1)+a(n-2)+1)) \\ Felix Fröhlich, Mar 26 2018

Formula

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

a(n) = 2*A000045(n) + A000045(n+2) - 1 = A000285(n)-1.

a(n) = 2*a(n-1) - a(n-3) for n>=3. - Ron Knott, Aug 25 2006

a(n) = (3*A000032(n) - A000045(n) - 2)/2. - Vladimir Joseph Stephan Orlovsky, Feb 02 2012

a(n) = 4*F(n) + F(n-1) - 1, where F = A000045. - Bruno Berselli, Feb 20 2017

a(n) = (-10 + (5-7*sqrt(5))*((1-sqrt(5))/2)^n + ((1+sqrt(5))/2)^n*(5+7*sqrt(5))) / 10. - Colin Barker, Feb 20 2017