%I #34 Sep 08 2022 08:45:11

%S 567,239841,114082668,55125843489,26697877691247,12934267027240356,

%T 6266540498895923463,3036106030479071781249,1470978970343729016987852,

%U 712682440446248640284336721,345291321126117622870522555983,167292036479044881831300837903684,81052212349412217472309893818152407

%N a(n) = S(4*n,4)/S(n,4) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

%H Vincenzo Librandi, <a href="/A087424/b087424.txt">Table of n, a(n) for n = 1..370</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (567,-40824,413343,-531441).

%F a(n) = (243+108*sqrt(5))^n+(243-108*sqrt(5))^n+((81+27*sqrt(5))/2)^n+((81-27*sqrt(5))/2)^n.

%F G.f.: -81*x*(26244*x^3-15309*x^2+1008*x-7) / ((729*x^2-486*x+1)*(729*x^2-81*x+1)). - _Colin Barker_, Dec 01 2012

%F a(n)/3^(3*n) = L(2*n)*L(4*n) = L(2*n) + L(6*n), L=A000032. - _Ehren Metcalfe_, Apr 21 2018

%F a(n) = 27^n*F(8*n)/F(2*n), F=A000045. - _Ehren Metcalfe_, Aug 03 2018

%t Table[(27^n Fibonacci[8 n] / Fibonacci[2 n]), {n, 15}] (* _Vincenzo Librandi_, Aug 04 2018 *)

%t LinearRecurrence[{567,-40824,413343,-531441},{567,239841,114082668,55125843489},20] (* _Harvey P. Dale_, Jun 23 2020 *)

%o (Magma) [27^n*Fibonacci(8*n)/Fibonacci(2*n): n in [1..15]]; // _Vincenzo Librandi_, Aug 04 2018

%Y Cf. A020876.

%K nonn,easy

%O 1,1

%A _Benoit Cloitre_, Oct 22 2003