OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).
FORMULA
a(2n-1) = ceiling(1/(4/(Fibonacci(6n)*sqrt(5)-Lucas(6n)+2)-2)), a(2n) = ceiling(1/(1-2/(Fibonacci(6n+3)*sqrt(5)-Lucas(6n+3)+2))-2). - Thomas Baruchel
G.f.: (4+9*x+4*x^2-x^3)/(1-18*x^2+x^4).
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0 :
a0(n) = -((2+sqrt(5))/(9+4*sqrt(5))^n)+(-2+sqrt(5))*(9+4*sqrt(5))^n.
a1(n) = (1/(9+4*sqrt(5))^n+(9+4*sqrt(5))^n)/2. (End)
MATHEMATICA
Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[20], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
Numerator[Convergents[Sqrt[20], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((2+Sqrt[5])/(9+4*Sqrt[5])^n)+(-2+Sqrt[5])*(9+4*Sqrt[5])^n //Simplify
a1[n_] := (1/(9+4*Sqrt[5])^n+(9+4*Sqrt[5])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
CROSSREFS
KEYWORD
nonn,cofr,frac,easy
AUTHOR
STATUS
approved