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A041030 Numerators of continued fraction convergents to sqrt(20).

%I A041030 #34 Jul 14 2015 16:51:32
%S A041030 4,9,76,161,1364,2889,24476,51841,439204,930249,7881196,16692641,
%T A041030 141422324,299537289,2537720636,5374978561,45537549124,96450076809,
%U A041030 817138163596,1730726404001,14662949395604
%N A041030 Numerators of continued fraction convergents to sqrt(20).
%H A041030 Vincenzo Librandi, <a href="/A041030/b041030.txt">Table of n, a(n) for n = 0..200</a>
%H A041030 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,18,0,-1).
%F A041030 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_
%F A041030 G.f.: (4+9*x+4*x^2-x^3)/(1-18*x^2+x^4).
%F A041030 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041030 Interspersion of 2 sequences [a0(n),a1(n)] for n>0 :
%F A041030 a0(n) = -((2+sqrt(5))/(9+4*sqrt(5))^n)+(-2+sqrt(5))*(9+4*sqrt(5))^n.
%F A041030 a1(n) = (1/(9+4*sqrt(5))^n+(9+4*sqrt(5))^n)/2. (End)
%t A041030 Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[20],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 17 2011 *)
%t A041030 Numerator[Convergents[Sqrt[20], 30]] (* _Vincenzo Librandi_, Oct 28 2013 *)
%t A041030 a0[n_] := -((2+Sqrt[5])/(9+4*Sqrt[5])^n)+(-2+Sqrt[5])*(9+4*Sqrt[5])^n //Simplify
%t A041030 a1[n_] := (1/(9+4*Sqrt[5])^n+(9+4*Sqrt[5])^n)/2 // Simplify
%t A041030 Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* _Gerry Martens_, Jul 11 2015 *)
%Y A041030 Cf. A010476, A041031, A087953.
%K A041030 nonn,cofr,frac,easy
%O A041030 0,1
%A A041030 _N. J. A. Sloane_