%I #43 Oct 09 2023 09:26:28

%S 1,2,13,89,610,4181,28657,196418,1346269,9227465,63245986,433494437,

%T 2971215073,20365011074,139583862445,956722026041,6557470319842,

%U 44945570212853,308061521170129,2111485077978050,14472334024676221

%N a(n) = 7*a(n-1) - a(n-2) for n>1, a(0)=1, a(1)=2.

%H G. C. Greubel, <a href="/A172968/b172968.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).

%F a(n) = (1/10)*((5+sqrt(5))*((7-3*sqrt(5))/2)^n + ((5-sqrt(5))*((7+3*sqrt(5))/2)^n)).

%F a(n) = sqrt(1 - 2*F(2n+1)*F(2n+2) + 5*(F(2n+1)*F(2n+2))^2), where F = A000045.

%F a(n) = sqrt(1 - 2*A081016(n) + 5*A081016(n)^2).

%F a(n) = A033891(n-1), n>0. - _R. J. Mathar_, Feb 08 2010

%F a(n) = (Lucas(4*n) - Fibonacci(4*n))/2, where Lucas = A000032. - _Gary Detlefs_, Nov 28 2010

%F G.f.: (1 - 5*x)/(1 - 7*x + x^2). - _Bruno Berselli_, Mar 29 2016

%F a(n) = Fibonacci(4*n-1). - _G. C. Greubel_, Jul 15 2019

%F a(n) = (a(n-1)^2 + 9)/a(n-2). - _Klaus Purath_, Aug 30 2020

%p with(combinat):F:= n-> fibonacci(n):L:=n-> 2*F(n+1)-F(n):

%p seq(1/2*(L(4*n)-F(4*n)), n=0..20); # _Gary Detlefs_, Nov 28 2010

%t Table[Sqrt[1-2m+5m^2]/.m ->Fibonacci[2n+1]Fibonacci[2n+2], {n, -1, 30}]

%t CoefficientList[Series[(1-5x)/(1-7x+x^2), {x, 0, 30}], x] (* _Michael De Vlieger_, Mar 29 2016 *)

%t Fibonacci[4*Range[0, 30] -1] (* _G. C. Greubel_, Jul 15 2019 *)

%o (Magma) [n le 2 select n else 7*Self(n-1)-Self(n-2): n in [1..30]]; // _Bruno Berselli_, Mar 29 2016

%o (PARI) x='x+O('x^30); Vec((1-5*x)/(1-7*x+x^2)) \\ _Altug Alkan_, Mar 29 2016

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

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

%Y Cf. A000032, A000045, A081016.

%Y Essentially the same as A033891.

%K nonn,easy

%O 0,2

%A _Artur Jasinski_, Feb 06 2010