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A041042 Numerators of continued fraction convergents to sqrt(27).

%I A041042 #38 Dec 12 2015 08:25:35
%S A041042 5,26,265,1351,13775,70226,716035,3650401,37220045,189750626,
%T A041042 1934726305,9863382151,100568547815,512706121226,5227629760075,
%U A041042 26650854921601,271736178976085,1385331749802026,14125053676996345
%N A041042 Numerators of continued fraction convergents to sqrt(27).
%C A041042 Subset of |A002316| (conjectured).
%H A041042 Vincenzo Librandi, <a href="/A041042/b041042.txt">Table of n, a(n) for n = 0..200</a>
%H A041042 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,52,0,-1).
%F A041042 G.f.: (-x^3+5x^2+26x+5)/(x^4-52x^2+1).
%F A041042 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041042 Interspersion of 2 sequences [a0(n),a1(n)]:
%F A041042 a0(n) = ((-5-3*sqrt(3))/(26+15*sqrt(3))^n+(-5+3*sqrt(3))*(26+15*sqrt(3))^n)/2.
%F A041042 a1(n) = (1/(26+15*sqrt(3))^n+(26+15*sqrt(3))^n)/2. (End)
%t A041042 Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[27],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2011*)
%t A041042 Numerator/@Convergents[Sqrt[27],20] (* _Harvey P. Dale_, Jul 21 2011 *)
%t A041042 CoefficientList[Series[(- x^3 + 5 x^2 + 26 x + 5)/(x^4 - 52 x^2 + 1), {x, 0, 30}], x]  (* _Vincenzo Librandi_, Oct 28 2013 *)
%t A041042 a0[n_] := (-5-3*Sqrt[3]+(-5+3*Sqrt[3])*(26+15*Sqrt[3])^(2*n))/(2*(26+15*Sqrt[3])^n) // Simplify
%t A041042 a1[n_] := (1+(26+15*Sqrt[3])^(2*n))/(2*(26+15*Sqrt[3])^n) //  Simplify
%t A041042 Flatten[MapIndexed[{a0[#], a1[#]}&,Range[10]]] (* _Gerry Martens_, Jul 10 2015 *)
%t A041042 LinearRecurrence[{0,52,0,-1},{5,26,265,1351},30] (* _Harvey P. Dale_, Dec 12 2015 *)
%Y A041042 Cf. A010482, A041043, A002316.
%K A041042 nonn,cofr,frac,easy
%O A041042 0,1
%A A041042 _N. J. A. Sloane_