A041030 Numerators of continued fraction convergents to sqrt(20).
4, 9, 76, 161, 1364, 2889, 24476, 51841, 439204, 930249, 7881196, 16692641, 141422324, 299537289, 2537720636, 5374978561, 45537549124, 96450076809, 817138163596, 1730726404001, 14662949395604
Offset: 0
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).
Programs
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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 *)
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)