A081011 - OEIS

%I #46 Nov 28 2024 14:43:40

%S 4,15,91,612,4183,28659,196420,1346271,9227467,63245988,433494439,

%T 2971215075,20365011076,139583862447,956722026043,6557470319844,

%U 44945570212855,308061521170131,2111485077978052,14472334024676223

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

%C For n>0, a(n) is the area of the trapezoid defined by the four points (F(n+1), F(n+2)), (F(n+2), F(n+1)), (F(n+3), F(n+4)), and (F(n+4), F(n+3)) where F(n) = A000045(n). - _J. M. Bergot_, May 14 2014

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

%H G. C. Greubel, <a href="/A081011/b081011.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 G.f.: (4-17*x+3*x^2)/((1-x)*(1-7*x+x^2)). - _Colin Barker_, Jun 22 2012

%F Product_{n>=0} (1 - 1/a(n)) = (3-phi)/2 = A187798. - _Amiram Eldar_, Nov 28 2024

%p with(combinat) for n from 0 to 30 do printf(`%d,`,fibonacci(4*n+3)+2) od # _James A. Sellers_, Mar 03 2003

%t Table[Fibonacci[4n+3] +2, {n,0,30}] (* or *)

%t Table[Fibonacci[2n+3]*LucasL[2n], {n, 0, 30}] (* _Alonso del Arte_, Apr 18 2011 *)

%t LinearRecurrence[{8,-8,1},{4,15,91},30] (* _Harvey P. Dale_, Apr 22 2017 *)

%o (Magma) [Fibonacci(4*n+3)+2: n in [0..30]]; // _Vincenzo Librandi_, Apr 18 2011

%o (PARI) vector(30, n, n--; fibonacci(4*n+3)+2) \\ _G. C. Greubel_, Jul 14 2019

%o (Sage) [fibonacci(4*n+3)+2 for n in (0..30)] # _G. C. Greubel_, Jul 14 2019

%o (GAP) List([0..30], n-> Fibonacci(4*n+3) -2); # _G. C. Greubel_, Jul 14 2019

%Y Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers), A187798.

%K nonn,easy

%O 0,1

%A _R. K. Guy_, Mar 01 2003

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