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A041047 Denominators of continued fraction convergents to sqrt(29).

%I A041047 #30 Jul 09 2025 00:11:41
%S A041047 1,2,3,5,13,135,283,418,701,1820,18901,39622,58523,98145,254813,
%T A041047 2646275,5547363,8193638,13741001,35675640,370497401,776670442,
%U A041047 1147167843,1923838285,4994844413,51872282415
%N A041047 Denominators of continued fraction convergents to sqrt(29).
%C A041047 The terms of this sequence can be constructed with the terms of sequence A052918.
%C A041047 For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484. - _Johannes W. Meijer_, Jun 12 2010
%H A041047 Vincenzo Librandi, <a href="/A041047/b041047.txt">Table of n, a(n) for n = 0..200</a>
%H A041047 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,140,0,0,0,0,1).
%F A041047 a(5*n) = A052918(3*n), a(5*n+1) = (A052918(3*n+1) - A052918(3*n))/2, a(5*n+2) = (A052918(3*n+1) + A052918(3*n))/2, a(5*n+3) = A052918(3*n+1) and a(5*n+4) = A052918(3*n+2)/2. - _Johannes W. Meijer_, Jun 12 2010
%F A041047 G.f.: (1 + 2*x + 3*x^2 + 5*x^3 + 13*x^4 - 5*x^5 + 3*x^6 - 2*x^7 + x^8)/(1 - 140*x^5 - x^10). - _Peter J. C. Moses_, Jul 29 2013
%F A041047 a(n) = 140*a(n-5) + a(n-10). - _Vincenzo Librandi_, Dec 10 2013
%t A041047 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 18 2011 *)
%t A041047 Denominator[Convergents[Sqrt[29], 30]] (* _Vincenzo Librandi_, Dec 10 2013 *)
%o A041047 (Magma) I:=[1, 2, 3, 5, 13, 135, 283, 418, 701, 1820]; [n le 10 select I[n] else 140*Self(n-5)+Self(n-10): n in [1..50]]; // _Vincenzo Librandi_, Dec 10 2013
%Y A041047 Cf. A041046.
%Y A041047 Cf. A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.
%K A041047 nonn,cofr,frac,easy
%O A041047 0,2
%A A041047 _N. J. A. Sloane_