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A041127 Denominators of continued fraction convergents to sqrt(72).

%I A041127 #28 Sep 08 2022 08:44:54
%S A041127 1,2,33,68,1121,2310,38081,78472,1293633,2665738,43945441,90556620,
%T A041127 1492851361,3076259342,50713000833,104502261008,1722749176961,
%U A041127 3550000614930,58522759015841,120595518646612,1988051057361633,4096697633369878,67535213191279681
%N A041127 Denominators of continued fraction convergents to sqrt(72).
%H A041127 Vincenzo Librandi, <a href="/A041127/b041127.txt">Table of n, a(n) for n = 0..200</a>
%H A041127 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,34,0,-1).
%F A041127 G.f.: -(x^2-2*x-1) / ((x^2-6*x+1)*(x^2+6*x+1)). - _Colin Barker_, Nov 13 2013
%F A041127 a(n) = 34*a(n-2) - a(n-4). - _Vincenzo Librandi_, Dec 11 2013
%F A041127 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041127 Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
%F A041127 a0(n) = ((3+2*sqrt(2))/(17+12*sqrt(2))^n+(3-2*sqrt(2))*(17+12*sqrt(2))^n)/6.
%F A041127 a1(n) = (-1/(17+12*sqrt(2))^n+(17+12*sqrt(2))^n)/(12*sqrt(2)). (End)
%t A041127 Denominator/@Convergents[Sqrt[72], 50] (* _Vladimir Joseph Stephan Orlovsky_, Jul 05 2011 *)
%t A041127 CoefficientList[Series[(1 + 2 x - x^2)/(x^4 - 34 x^2 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 11 2013 *)
%t A041127 a0[n_] := ((3+2*Sqrt[2])/(17+12*Sqrt[2])^n+(3-2*Sqrt[2])*(17+ 12*Sqrt[2])^n)/6 // Simplify
%t A041127 a1[n_] := (-1/(17+12*Sqrt[2])^n+(17+12*Sqrt[2])^n)/(12*Sqrt[2]) // FullSimplify
%t A041127 Flatten[MapIndexed[{a0[#],a1[#]}&,Range[20]]] (* _Gerry Martens_, Jul 10 2015 *)
%o A041127 (PARI) a(n)=my(v=contfrac(sqrt(72),n),s=v[n]);forstep(k=n-1,1,-1,s=v[k]+1/s);denominator(s) \\ _Charles R Greathouse IV_, Jul 05 2011
%o A041127 (Magma) I:=[1, 2, 33, 68]; [n le 4 select I[n] else 34*Self(n-2)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Dec 11 2013
%Y A041127 Cf. A041126, A040063, A020829, A010524.
%K A041127 nonn,frac,easy
%O A041127 0,2
%A A041127 _N. J. A. Sloane_