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A041427 Denominators of continued fraction convergents to sqrt(229).

%I A041427 #30 Jul 09 2025 00:48:47
%S A041427 1,7,8,15,113,3405,23948,27353,51301,386460,11645101,81902167,
%T A041427 93547268,175449435,1321693313,39826248825,280105435088,319931683913,
%U A041427 600037119001,4520191516920,136205782626601,957960669903127,1094166452529728,2052127122432855
%N A041427 Denominators of continued fraction convergents to sqrt(229).
%C A041427 The a(n) terms of this sequence can be constructed with the terms of sequence A154597. For the terms of the periodical sequence of the continued fraction for sqrt(229) see A040213. We observe that its period is five. - _Johannes W. Meijer_, Jun 12 2010
%H A041427 Vincenzo Librandi, <a href="/A041427/b041427.txt">Table of n, a(n) for n = 0..200</a>
%H A041427 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 3420, 0, 0, 0, 0, 1).
%F A041427 a(5*n) = A154597(3*n+1), a(5*n+1) = (A154597(3*n+2) - A154597(3*n+1))/2, a(5*n+2) = (A154597(3*n+2) + A154597(3*n+1))/2, a(5*n+3) = A154597(3*n+2) and a(5*n+4) = A154597(3*n+3)/2. - _Johannes W. Meijer_, Jun 12 2010
%F A041427 G.f.: -(x^8 -7*x^7 +8*x^6 -15*x^5 +113*x^4 +15*x^3 +8*x^2 +7*x +1) / (x^10 +3420*x^5 -1). - _Colin Barker_, Nov 12 2013
%F A041427 a(n) = 3420*a(n-5) + a(n-10) for n>9. - _Vincenzo Librandi_, Dec 17 2013
%t A041427 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[229], n]]], {n, 1, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 23 2011 *)
%t A041427 Denominator[Convergents[Sqrt[229], 30]] (* _Vincenzo Librandi_, Dec 17 2013 *)
%t A041427 LinearRecurrence[{0,0,0,0,3420,0,0,0,0,1},{1,7,8,15,113,3405,23948,27353,51301,386460},30] (* _Harvey P. Dale_, Oct 14 2020 *)
%o A041427 (Magma) I:=[1,7,8,15,113,3405,23948,27353,51301,386460]; [n le 10 select I[n] else 3420*Self(n-5)+Self(n-10): n in [1..40]]; // _Vincenzo Librandi_, Dec 17 2013
%Y A041427 Cf. A041426, A166125, A040213, A041019, A041047, A041091, A041151, A041227, A041319, A041427 and A041551.
%K A041427 nonn,easy,frac
%O A041427 0,2
%A A041427 _N. J. A. Sloane_