A022116 Fibonacci sequence beginning 2, 13.

2, 13, 15, 28, 43, 71, 114, 185, 299, 484, 783, 1267, 2050, 3317, 5367, 8684, 14051, 22735, 36786, 59521, 96307, 155828, 252135, 407963, 660098, 1068061, 1728159, 2796220, 4524379, 7320599, 11844978, 19165577, 31010555, 50176132, 81186687, 131362819

Offset: 0

Author

N. J. A. Sloane, Jun 14 1998

Keywords

• nonn
• easy

Links

• Ivan Panchenko, Table of n, a(n) for n = 0..1000
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (1,1).

Crossrefs

Cf. A000032, A000045.

Programs

Magma

a:=[2,13]; [n le 2 select a[n] else Self(n-1)+Self(n-2): n in [1..36]]; // Marius A. Burtea, Feb 11 2020

Magma

R:=PowerSeriesRing(Integers(), 36); Coefficients(R!( (2+11*x)/(1-x-x^2))); // Marius A. Burtea, Feb 11 2020

Maple

seq( 2*fibonacci(n+2) +9*fibonacci(n), n=0..40); # G. C. Greubel, Feb 12 2020

Mathematica

CoefficientList[Series[(2+11x)/(1-x-x^2), {x, 0, 40}], x] (* Wesley Ivan Hurt, Jun 15 2014 *)
LinearRecurrence[{1,1},{2,13},50] (* Harvey P. Dale, Jun 20 2017 *)

PARI

vector(41, n, my(m=n-1, f=fibonacci); 2*f(m+2) + 9*f(m) ) \\ G. C. Greubel, Feb 12 2020

Sage

[2*fibonacci(n+2) + 9*fibonacci(n) for n in (0..40)] # G. C. Greubel, Feb 12 2020

Formula

G.f.: (2 + 11*x)/(1-x-x^2). - Philippe Deléham, Nov 19 2008

a(n) = 4*Fibonacci(n+2) + 3*Fibonacci(n+3) - 4*Lucas(n). - Lechoslaw Ratajczak, Feb 10 2020

E.g.f.: (2/5)*exp(x/2)*(5*cosh(sqrt(5)*x/2) + 12*sqrt(5)*sinh(sqrt(5)*x/2)). - Stefano Spezia, Feb 11 2020

a(n) = 2*Fibonacci(n+2) + 9*Fibonacci(n). - G. C. Greubel, Feb 12 2020