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A041015 Denominators of continued fraction convergents to sqrt(11).

%I A041015 #47 Feb 05 2022 16:10:16
%S A041015 1,3,19,60,379,1197,7561,23880,150841,476403,3009259,9504180,60034339,
%T A041015 189607197,1197677521,3782639760,23893516081,75463188003,476672644099,
%U A041015 1505481120300,9509559365899,30034159217997
%N A041015 Denominators of continued fraction convergents to sqrt(11).
%C A041015 Sqrt(11) = 3 + continued fraction [3, 6, 3, 6, 3, 6, ...] = 6/2 + 6/19 + 6/(19*379) + 6/(379*7561) + ... - _Gary W. Adamson_, Dec 21 2007
%C A041015 Let X = the 2 X 2 matrix [1, 6; 3, 19], then X^n * [1, 0] = [a(n+1), a(n+2)]; e.g., X^3 * [1, 0] = [379, 1197] = [a(4), a(5)]. - _Gary W. Adamson_, Dec 21 2007
%H A041015 Vincenzo Librandi, <a href="/A041015/b041015.txt">Table of n, a(n) for n = 0..200</a>
%H A041015 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,20,0,-1).
%F A041015 G.f.: (1+3*x-x^2)/(1-20*x^2+x^4). - _Colin Barker_, Dec 31 2011
%F A041015 From _Gerry Martens_, Jul 11 2015: (Start)
%F A041015 Interspersion of 2 sequences [a0(n),a1(n)]:
%F A041015 a0(n) = ((11+3*sqrt(11))/(10+3*sqrt(11))^n + (11-3*sqrt(11))*(10+3*sqrt(11))^n)/22.
%F A041015 a1(n) = 3*Sum_{i=1..n} a0(i). (End)
%t A041015 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[11],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%t A041015 a0[n_] := (11+3*Sqrt[11]+(11-3*Sqrt[11])*(10+3*Sqrt[11])^(2*n))/(22*(10+3*Sqrt[11])^n) // Simplify
%t A041015 a1[n_] := 3*Sum[a0[i], {i, 1, n}]
%t A041015 Flatten[MapIndexed[{a0[#], a1[#]}&,Range[11]]] (* _Gerry Martens_, Jul 10 2015 *)
%Y A041015 Cf. A010468, A041014.
%K A041015 nonn,cofr,frac,easy
%O A041015 0,2
%A A041015 _N. J. A. Sloane_