OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-1).
FORMULA
a(n) = (1/10)*((5+sqrt(5))*((7-3*sqrt(5))/2)^n + ((5-sqrt(5))*((7+3*sqrt(5))/2)^n)).
a(n) = sqrt(1 - 2*F(2n+1)*F(2n+2) + 5*(F(2n+1)*F(2n+2))^2), where F = A000045.
a(n) = A033891(n-1), n>0. - R. J. Mathar, Feb 08 2010
a(n) = (Lucas(4*n) - Fibonacci(4*n))/2, where Lucas = A000032. - Gary Detlefs, Nov 28 2010
G.f.: (1 - 5*x)/(1 - 7*x + x^2). - Bruno Berselli, Mar 29 2016
a(n) = Fibonacci(4*n-1). - G. C. Greubel, Jul 15 2019
a(n) = (a(n-1)^2 + 9)/a(n-2). - Klaus Purath, Aug 30 2020
MAPLE
with(combinat):F:= n-> fibonacci(n):L:=n-> 2*F(n+1)-F(n):
seq(1/2*(L(4*n)-F(4*n)), n=0..20); # Gary Detlefs, Nov 28 2010
MATHEMATICA
Table[Sqrt[1-2m+5m^2]/.m ->Fibonacci[2n+1]Fibonacci[2n+2], {n, -1, 30}]
CoefficientList[Series[(1-5x)/(1-7x+x^2), {x, 0, 30}], x] (* Michael De Vlieger, Mar 29 2016 *)
Fibonacci[4*Range[0, 30] -1] (* G. C. Greubel, Jul 15 2019 *)
PROG
(Magma) [n le 2 select n else 7*Self(n-1)-Self(n-2): n in [1..30]]; // Bruno Berselli, Mar 29 2016
(PARI) x='x+O('x^30); Vec((1-5*x)/(1-7*x+x^2)) \\ Altug Alkan, Mar 29 2016
(Sage) [fibonacci(4*n-1) for n in (0..30)] # G. C. Greubel, Jul 15 2019
(GAP) List([0..30], n-> Fibonacci(4*n-1)); # G. C. Greubel, Jul 15 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Artur Jasinski, Feb 06 2010
STATUS
approved