%I #68 Jun 27 2022 11:08:18

%S 3,8,11,19,30,49,79,128,207,335,542,877,1419,2296,3715,6011,9726,

%T 15737,25463,41200,66663,107863,174526,282389,456915,739304,1196219,

%U 1935523,3131742,5067265,8199007,13266272,21465279,34731551,56196830,90928381,147125211

%N Fibonacci sequence beginning 3, 8.

%C From _Wajdi Maaloul_, Jun 20 2022: (Start)

%C a(n) is the number of ways to tile the figure below with squares and dominoes (a strip of length n+2 that contains two vertical strip of height 2 in its first and third tiles).

%C _ _

%C |_|_|_|_____ _

%C |_|_|_|_|_|_|...|_|

%C (End)

%H Vincenzo Librandi, <a href="/A022121/b022121.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.: (3+5*x)/(1-x-x^2). - _Philippe Deléham_, Nov 19 2008

%F a(n) = F(n+2) + 2*L(n+1), where F(n) and L(n) are the n-th Fibonacci and Lucas number, respectively. - _Rigoberto Florez_, Jan 29 2020

%F E.g.f.: exp(-2*x/(1+sqrt(5)))*(-2*(5+6*sqrt(5))+(55+27*sqrt(5))*exp(sqrt(5)*x))/(5*(3+sqrt(5))). - _Stefano Spezia_, Jan 29 2020 after _Rigoberto Florez_

%F a(n) = Lucas(n+3) + Fibonacci(n-2). - _Greg Dresden_ and Steve Warren, Feb 28 2022

%t LinearRecurrence[{1, 1}, {3, 8}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Sep 17 2008 *)

%t Table[2 LucasL[n+1] + Fibonacci[n+2], {n,0,40}] (* _Rigoberto Florez_, Jan 29 2020 *)

%o (Magma) a0:=3; a1:=8; [GeneralizedFibonacciNumber(a0, a1, n): n in [0..40]]; // _Vincenzo Librandi_, Jan 25 2017

%o (PARI) apply( {A022121(n)=[3,8]*([0,1;1,1]^n)[,1]}, [0..30]) \\ _M. F. Hasler_, Feb 28 2020

%Y Cf. A000032, A000045.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_