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A041063 Denominators of continued fraction convergents to sqrt(38).

%I A041063 #29 Feb 29 2024 19:20:28
%S A041063 1,6,73,444,5401,32850,399601,2430456,29565073,179820894,2187415801,
%T A041063 13304315700,161839204201,984339540906,11973913695073,72827821711344,
%U A041063 885907774231201,5388274467098550,65545201379413801,398659482743581356,4849458994302390073
%N A041063 Denominators of continued fraction convergents to sqrt(38).
%H A041063 Vincenzo Librandi, <a href="/A041063/b041063.txt">Table of n, a(n) for n = 0..200</a>
%H A041063 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,74,0,-1).
%F A041063 G.f.: -(x^2-6*x-1) / (x^4-74*x^2+1). - _Colin Barker_, Nov 12 2013
%F A041063 a(n) = 74*a(n-2) - a(n-4). - _Vincenzo Librandi_, Dec 10 2013
%F A041063 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041063 Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
%F A041063 a0(n) = ((38+6*sqrt(38))/(37+6*sqrt(38))^n+(38-6*sqrt(38))*(37+6*sqrt(38))^n)/76;
%F A041063 a1(n) = (-1/(37+6*sqrt(38))^n+(37+6*sqrt(38))^n)/(2*sqrt(38)). (End)
%t A041063 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[38], n]]], {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 21 2011*)
%t A041063 Denominator[Convergents[Sqrt[38], 30]] (* _Vincenzo Librandi_, Dec 10 2013 *)
%t A041063 a0[n_] := ((38+6*Sqrt[38])/(37+6*Sqrt[38])^n+(38-6*Sqrt[38])*(37+6*Sqrt[38])^n)/76 // Simplify
%t A041063 a1[n_] := (-1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/(2*Sqrt[38]) // FullSimplify
%t A041063 Flatten[MapIndexed[{a0[#],a1[#]}&, Range[20]]] (* _Gerry Martens_, Jul 11 2015 *)
%t A041063 LinearRecurrence[{0,74,0,-1},{1,6,73,444},30] (* _Harvey P. Dale_, Feb 29 2024 *)
%o A041063 (Magma) I:=[1, 6, 73, 444]; [n le 4 select I[n] else 74*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 10 2013
%Y A041063 Cf. A010492, A041062.
%K A041063 nonn,cofr,frac,easy
%O A041063 0,2
%A A041063 _N. J. A. Sloane_
%E A041063 More terms from _Colin Barker_, Nov 12 2013