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%I #60 Jan 02 2024 08:52:10
%S 0,11,87,608,4179,28655,196416,1346267,9227463,63245984,433494435,
%T 2971215071,20365011072,139583862443,956722026039,6557470319840,
%U 44945570212851,308061521170127,2111485077978048,14472334024676219
%N a(n) = Fibonacci(4*n+3) - 2, or Fibonacci(2*n)*Lucas(2*n+3).
%D Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H G. C. Greubel, <a href="/A081013/b081013.txt">Table of n, a(n) for n = 0..1000</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 From _R. J. Mathar_, Sep 03 2010: (Start)
%F G.f.: x*(11 - x)/((1-x)*(1-7*x+x^2)).
%F a(n) = A033891(n) - 2.
%F a(n+1) - a(n) = A056914(n+1), n>0. (End)
%F a(n) = 7*a(n-1) - a(n-2) + 10, n>=2. - _R. J. Mathar_, Nov 07 2015
%F From _Rigoberto Florez_, Apr 20 2019: (Start)
%F a(n) = Sum_{i=0..2n} F(i)*L(i+2), F(i) = A000045(i) and L(i) = A000032(i).
%F a(n) = (Sum_{i=1..2n-1} binomial(2n-1+i,2n-1-i)) - 1. (End)
%p with(combinat) for n from 0 to 40 do printf(`%d,`,fibonacci(4*n+3)-2) od # _James A. Sellers_, Mar 03 2003
%t LinearRecurrence[{8,-8,1},{0,11,87},40] (* _Harvey P. Dale_, Dec 05 2013 *)
%t Table[Fibonacci[2n] LucasL[2n+3], {n,1,40}] (* _Rigoberto Florez_, Apr 20 2019 *)
%t Table[Sum[Binomial[2n-1+i, 2n-1-i], {i, 1, 2n-1}]-1, {n, 1, 40}] (* _Rigoberto Florez_, Apr 20 2019 *)
%o (Magma) [Fibonacci(4*n+3)-2: n in [0..40]]; // _Vincenzo Librandi_, Apr 20 2011
%o (PARI) my(x='x+O('x^40)); concat([0], Vec(x*(11-x)/((1-x)*(1-7*x+x^2)))) \\ _G. C. Greubel_, Dec 24 2017
%o (PARI) vector(40, n, n--; fibonacci(4*n+3)-2) \\ _G. C. Greubel_, Jul 14 2019
%o (Magma) [Fibonacci(4*n+3)-2: n in [0..40]]; // _G. C. Greubel_, Jul 14 2019
%o (Sage) [fibonacci(4*n+3)-2 for n in (0..40)] # _G. C. Greubel_, Jul 14 2019
%o (GAP) List([0..40], n-> Fibonacci(4*n+3) -2); # _G. C. Greubel_, Jul 14 2019
%Y Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).
%K nonn,easy
%O 0,2
%A _R. K. Guy_, Mar 01 2003
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