A022408 a(n) = a(n-1) + a(n-2) + 1, with a(0)=3, a(1)=9.

3, 9, 13, 23, 37, 61, 99, 161, 261, 423, 685, 1109, 1795, 2905, 4701, 7607, 12309, 19917, 32227, 52145, 84373, 136519, 220893, 357413, 578307, 935721, 1514029, 2449751, 3963781, 6413533, 10377315, 16790849, 27168165, 43959015, 71127181, 115086197, 186213379, 301299577, 487512957

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Links

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

Crossrefs

Cf. A022382.

Programs

Magma

I:=[3,9,13]; [n le 3 select I[n] else 2*Self(n-1) - Self(n-3): n in [1..40]]; // G. C. Greubel, Mar 01 2018

Mathematica

CoefficientList[Series[(3+3*x-5*x^2)/((x-1)*(x^2+x-1)), {x, 0, 50}], x] (* or *) LinearRecurrence[{2,0,-1}, {3,9,13}, 50] (* G. C. Greubel, Mar 01 2018 *)
nxt[{a_,b_}]:={b,a+b+1}; NestList[nxt,{3,9},40][[;;,1]] (* Harvey P. Dale, Sep 13 2024 *)

PARI

x='x+O('x^40); Vec((3+3*x-5*x^2)/((x-1)*(x^2+x-1))) \\ G. C. Greubel, Mar 01 2018

Formula

From R. J. Mathar, Mar 11 2011: (Start)

a(n+1) - a(n) = A022382(n-1).

G.f.: ( 3+3*x-5*x^2 ) / ( (x-1)*(x^2+x-1) ). (End)

a(n) = 2*Lucas(n+1) + 2*Fibonacci(n+2) - 1. - Greg Dresden, Oct 10 2020

Extensions

Terms a(31) onward added by G. C. Greubel, Mar 01 2018