A087424 a(n) = S(4*n,4)/S(n,4) where S(n,m) = Sum_{k=0..n} binomial(n,k)*Fibonacci(m*k).

567, 239841, 114082668, 55125843489, 26697877691247, 12934267027240356, 6266540498895923463, 3036106030479071781249, 1470978970343729016987852, 712682440446248640284336721, 345291321126117622870522555983, 167292036479044881831300837903684, 81052212349412217472309893818152407

Offset: 1

Author

Benoit Cloitre, Oct 22 2003

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..370
• Index entries for linear recurrences with constant coefficients, signature (567,-40824,413343,-531441).

Crossrefs

Cf. A020876.

Programs

Magma

[27^n*Fibonacci(8*n)/Fibonacci(2*n): n in [1..15]]; // Vincenzo Librandi, Aug 04 2018

Mathematica

Table[(27^n Fibonacci[8 n] / Fibonacci[2 n]), {n, 15}] (* Vincenzo Librandi, Aug 04 2018 *)
LinearRecurrence[{567,-40824,413343,-531441},{567,239841,114082668,55125843489},20] (* Harvey P. Dale, Jun 23 2020 *)

Formula

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

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

a(n)/3^(3*n) = L(2*n)*L(4*n) = L(2*n) + L(6*n), L=A000032. - Ehren Metcalfe, Apr 21 2018

a(n) = 27^n*F(8*n)/F(2*n), F=A000045. - Ehren Metcalfe, Aug 03 2018