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A041117 Denominators of continued fraction convergents to sqrt(67).

%I A041117 #24 Jul 09 2025 00:17:33
%S A041117 1,5,11,16,27,205,232,437,1106,5967,96578,488857,1074292,1563149,
%T A041117 2637441,20025236,22662677,42687913,108038503,582880428,9434125351,
%U A041117 47753507183,104941139717,152694646900,257635786617,1956145153219,2213780939836,4169926093055
%N A041117 Denominators of continued fraction convergents to sqrt(67).
%H A041117 Vincenzo Librandi, <a href="/A041117/b041117.txt">Table of n, a(n) for n = 0..200</a>
%H A041117 <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,97684,0,0,0,0,0,0,0,0,0,-1).
%F A041117 G.f.: -(x^18 -5*x^17 +11*x^16 -16*x^15 +27*x^14 -205*x^13 +232*x^12 -437*x^11 +1106*x^10 -5967*x^9 -1106*x^8 -437*x^7 -232*x^6 -205*x^5 -27*x^4 -16*x^3 -11*x^2 -5*x -1) / (x^20 -97684*x^10 +1). - _Colin Barker_, Nov 13 2013
%F A041117 a(n) = 97684*a(n-10) - a(n-20). - _Vincenzo Librandi_, Dec 11 2013
%t A041117 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[67],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%t A041117 Denominator[Convergents[Sqrt[67], 30]] (* _Harvey P. Dale_, Oct 03 2012 *)
%t A041117 CoefficientList[Series[-(x^18 - 5 x^17 + 11 x^16 - 16 x^15 + 27 x^14 - 205 x^13 + 232 x^12 - 437 x^11 + 1106 x^10 - 5967 x^9 - 1106 x^8 - 437 x^7 - 232 x^6 - 205 x^5 - 27 x^4 - 16 x^3 - 11 x^2 - 5 x - 1)/(x^20 - 97684 x^10 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 11 2013 *)
%o A041117 (Magma) I:=[1, 5, 11, 16, 27, 205, 232, 437, 1106, 5967, 96578, 488857, 1074292, 1563149, 2637441, 20025236, 22662677, 42687913, 108038503, 582880428]; [n le 20 select I[n] else 97684*Self(n-10)-Self(n-20): n in [1..40]]; // _Vincenzo Librandi_, Dec 11 2013
%Y A041117 Cf. A041116, A010147, A020824, A010519.
%K A041117 nonn,cofr,frac,easy
%O A041117 0,2
%A A041117 _N. J. A. Sloane_
%E A041117 More terms from _Colin Barker_, Nov 13 2013