OFFSET
0,2
LINKS
Colin Barker, Table of n, a(n) for n = 0..700
R. S. Melham and A. G. Shannon, Inverse trigonometric and hyperbolic summation formulas involving generalized Fibonacci numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
Index entries for linear recurrences with constant coefficients, signature (17,17,-1).
FORMULA
a(n) = (1/5)*(Fibonacci(6*n+3) - 2*Fibonacci(6*n) - 2*(-1)^n). - Ralf Stephan, May 14 2004
G.f.: 4*(-x^2+x)/((1+x)*(1-18*x+x^2)). - Ralf Stephan, May 14 2004
a(n) = Fibonacci(3*n)^2. - Gary Detlefs, Nov 28 2010
a(n) = (-1)^(n+1)*(Fibonacci(n)*Fibonacci(7*n)-Fibonacci(4*n)^2). - Gary Detlefs, Nov 28 2010
a(n) = (-2*(-1)^n+(9+4*sqrt(5))^(-n)+(9+4*sqrt(5))^n)/5. - Colin Barker, Mar 04 2016
a(n) = A014445(n)^2. - Sean A. Irvine, Nov 18 2018
From Amiram Eldar, Jan 02 2026: (Start)
Sum_{n>=1} (-1)^(n+1) * arctanh(1/a(n)) = log(phi)/2 (A202541) (Melham and Shannon, 1995, p. 39, eq. (4.5)).
Product_{n>=1} (a(n)-(-1)^n)/(a(n)+(-1)^n) = phi (A001622) (Melham and Shannon, 1995, p. 39, eq. (4.6)). (End)
MATHEMATICA
(Table[Fibonacci@ n, {n, 0, 55}] /. n_ /; OddQ@ n -> Nothing)^2 (* or *)
CoefficientList[Series[4 (-x^2 + x)/((1 + x) (1 - 18 x + x^2)), {x, 0, 18}], x] (* Michael De Vlieger, Mar 04 2016 *)
LinearRecurrence[{17, 17, -1}, {0, 4, 64}, 20] (* Harvey P. Dale, Aug 02 2024 *)
PROG
(MuPAD) numlib::fibonacci(3*n)^2 $ n = 0..25; // Zerinvary Lajos, May 09 2008
(SageMath) [fibonacci(3*n)^2 for n in range(0, 17)] # Zerinvary Lajos, May 15 2009
(PARI) concat(0, Vec(4*x*(1-x)/((1+x)*(1-18*x+x^2)) + O(x^40))) \\ Colin Barker, Mar 04 2016
(Magma) [Fibonacci(3*n)^2: n in [0..20]]; // Vincenzo Librandi, Nov 19 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from James Sellers
STATUS
approved