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Fibonacci sequence beginning 2, 13.
3

%I #44 Sep 08 2022 08:44:45

%S 2,13,15,28,43,71,114,185,299,484,783,1267,2050,3317,5367,8684,14051,

%T 22735,36786,59521,96307,155828,252135,407963,660098,1068061,1728159,

%U 2796220,4524379,7320599,11844978,19165577,31010555,50176132,81186687,131362819

%N Fibonacci sequence beginning 2, 13.

%H Ivan Panchenko, <a href="/A022116/b022116.txt">Table of n, a(n) for n = 0..1000</a>

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,1).

%F G.f.: (2 + 11*x)/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008

%F a(n) = 4*Fibonacci(n+2) + 3*Fibonacci(n+3) - 4*Lucas(n). - _Lechoslaw Ratajczak_, Feb 10 2020

%F 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

%F a(n) = 2*Fibonacci(n+2) + 9*Fibonacci(n). - _G. C. Greubel_, Feb 12 2020

%p seq( 2*fibonacci(n+2) +9*fibonacci(n), n=0..40); # _G. C. Greubel_, Feb 12 2020

%t CoefficientList[Series[(2+11x)/(1-x-x^2), {x, 0, 40}], x] (* _Wesley Ivan Hurt_, Jun 15 2014 *)

%t LinearRecurrence[{1,1},{2,13},50] (* _Harvey P. Dale_, Jun 20 2017 *)

%o (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

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 36); Coefficients(R!( (2+11*x)/(1-x-x^2))); // _Marius A. Burtea_, Feb 11 2020

%o (PARI) vector(41, n, my(m=n-1, f=fibonacci); 2*f(m+2) + 9*f(m) ) \\ _G. C. Greubel_, Feb 12 2020

%o (Sage) [2*fibonacci(n+2) + 9*fibonacci(n) for n in (0..40)] # _G. C. Greubel_, Feb 12 2020

%Y Cf. A000032, A000045.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, Jun 14 1998