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Denominators of continued fraction convergents to sqrt(106).
2

%I #27 Sep 08 2022 08:44:54

%S 1,3,7,10,17,27,44,115,389,7895,24074,56043,80117,136160,216277,

%T 352437,921151,3115890,63238951,192832743,448904437,641737180,

%U 1090641617,1732378797,2823020414,7378419625,24958279289,506544005405,1544590295504,3595724596413

%N Denominators of continued fraction convergents to sqrt(106).

%H Vincenzo Librandi, <a href="/A041191/b041191.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,8010,0,0,0,0,0,0,0,0,1).

%F G.f.: -(x^16 -3*x^15 +7*x^14 -10*x^13 +17*x^12 -27*x^11 +44*x^10 -115*x^9 +389*x^8 +115*x^7 +44*x^6 +27*x^5 +17*x^4 +10*x^3 +7*x^2 +3*x +1) / (x^18 +8010*x^9 -1). - _Colin Barker_, Nov 14 2013

%F a(n) = 8010*a(n-9) + a(n-18). - _Vincenzo Librandi_, Dec 12 2013

%t Denominator[Convergents[Sqrt[106], 30]] (* _Vincenzo Librandi_, Dec 12 2013 *)

%o (Magma) I:=[1,3,7,10,17,27,44,115,389,7895,24074,56043,80117,136160, 216277,352437,921151,3115890]; [n le 18 select I[n] else 8010*Self(n-9)+Self(n-18): n in [1..40]]; // _Vincenzo Librandi_, Dec 12 2013

%o (Python)

%o from sympy import sqrt

%o from sympy.ntheory.continued_fraction import *

%o def aupton(terms):

%o g = continued_fraction_convergents(continued_fraction_iterator(sqrt(106)))

%o return [next(g).denominator() for n in range(terms)]

%o print(aupton(30)) # _Michael S. Branicky_, Oct 31 2021

%Y Cf. A041190, A010172.

%K nonn,frac,easy

%O 0,2

%A _N. J. A. Sloane_.

%E More terms from _Colin Barker_, Nov 14 2013