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a(n) = Fibonacci(4n+3) + 1, or Fibonacci(2n+1)*Lucas(2n+2).
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%I #28 Jan 01 2024 11:07:34

%S 3,14,90,611,4182,28658,196419,1346270,9227466,63245987,433494438,

%T 2971215074,20365011075,139583862446,956722026042,6557470319843,

%U 44945570212854,308061521170130,2111485077978051,14472334024676222,99194853094755498,679891637638612259

%N a(n) = Fibonacci(4n+3) + 1, or Fibonacci(2n+1)*Lucas(2n+2).

%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

%H Nathaniel Johnston, <a href="/A081005/b081005.txt">Table of n, a(n) for n = 0..500</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).

%F a(n) = 8*a(n-1) - 8*a(n-2) + a(n-3).

%F G.f.: (3-10*x+2*x^2)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 24 2012

%p with(combinat): for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+3)+1) od: # _James A. Sellers_, Mar 03 2003

%t Fibonacci[4Range[0,30]+3]+1 (* or *) LinearRecurrence[{8,-8,1}, {3,14,90}, 30] (* _Harvey P. Dale_, Jan 02 2013 *)

%o (Magma) [Fibonacci(4*n+3)+1: n in [0..30]]; // _Vincenzo Librandi_, Apr 15 2011

%o (PARI) vector(30, n, n--; fibonacci(4*n+3)+1) \\ _G. C. Greubel_, Jul 15 2019

%o (Sage) [fibonacci(4*n+3)+1 for n in (0..30)] # _G. C. Greubel_, Jul 15 2019

%o (GAP) List([0..30], n-> Fibonacci(4*n+3)+1); # _G. C. Greubel_, Jul 15 2019

%Y Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).

%K nonn,easy

%O 0,1

%A _R. K. Guy_, Mar 01 2003

%E More terms from _James A. Sellers_, Mar 03 2003