login
Denominators of continued fraction convergents to sqrt(86).
2

%I #20 Sep 08 2022 08:44:54

%S 1,3,4,7,11,95,106,201,307,1122,20503,62631,83134,145765,228899,

%T 1976957,2205856,4182813,6388669,23348820,426667429,1303351107,

%U 1730018536,3033369643,4763388179,41140475075,45903863254,87044338329,132948201583,485888943078

%N Denominators of continued fraction convergents to sqrt(86).

%H Vincenzo Librandi, <a href="/A041153/b041153.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,20810,0,0,0,0,0,0,0,0,0,-1).

%F G.f.: -(x^18 -3*x^17 +4*x^16 -7*x^15 +11*x^14 -95*x^13 +106*x^12 -201*x^11 +307*x^10 -1122*x^9 -307*x^8 -201*x^7 -106*x^6 -95*x^5 -11*x^4 -7*x^3 -4*x^2 -3*x-1) / (x^20-20810*x^10+1). - _Colin Barker_, Nov 13 2013

%F a(n) = 20810*a(n-10) - a(n-20). - _Vincenzo Librandi_, Dec 12 2013

%t Denominator/@Convergents[Sqrt[86], 50] (* _Vladimir Joseph Stephan Orlovsky_, Jul 05 2011 *)

%t CoefficientList[Series[-(x^18 - 3 x^17 + 4 x^16 - 7 x^15 + 11 x^14 - 95 x^13 + 106 x^12 - 201 x^11 + 307 x^10 - 1122 x^9 - 307 x^8 - 201 x^7 - 106 x^6 - 95 x^5 - 11 x^4 - 7 x^3 - 4 x^2 - 3 x - 1)/(x^20 - 20810 x^10 + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 12 2013 *)

%o (Magma) I:=[1, 3, 4, 7, 11, 95, 106, 201, 307, 1122, 20503, 62631, 83134, 145765, 228899, 1976957, 2205856, 4182813, 6388669, 23348820]; [n le 20 select I[n] else 20810*Self(n-10)-Self(n-20): n in [1..40]]; // _Vincenzo Librandi_, Dec 12 2013

%Y Cf. A041152, A010159, A020843, A010537.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_.