cp's OEIS Frontend

A041007 Denominators of continued fraction convergents to sqrt(6).

%I A041007 #56 Oct 31 2024 01:52:13
%S A041007 1,2,9,20,89,198,881,1960,8721,19402,86329,192060,854569,1901198,
%T A041007 8459361,18819920,83739041,186298002,828931049,1844160100,8205571449,
%U A041007 18255302998,81226783441,180708869880
%N A041007 Denominators of continued fraction convergents to sqrt(6).
%C A041007 sqrt(6) = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721), ...; where sqrt(6) = 2.4494897427... and the sum of the first 5 terms of this series = 2.449489737... - _Gary W. Adamson_, Dec 21 2007
%C A041007 sqrt(6) = 2 + continued fraction [2, 4, 2, 4, 2, 4, ...] = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721) + ... - _Gary W. Adamson_, Dec 21 2007
%C A041007 Interspersion of 2 sequences, A072256 and 2*A004189. - _Gerry Martens_, Jun 10 2015
%C A041007 For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [2, 4, 2, 4, ...] and 1's along the superdiagonal and the subdiagonal. - _Rogério Serôdio_, Apr 01 2018
%H A041007 Vincenzo Librandi, <a href="/A041007/b041007.txt">Table of n, a(n) for n = 0..1000</a>
%H A041007 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,10,0,-1).
%F A041007 G.f.: (1+2*x-x^2)/(1-10*x^2+x^4). - _Colin Barker_, Dec 31 2011
%F A041007 From _Rogério Serôdio_, Apr 01 2018: (Start)
%F A041007 Recurrence formula: a(n) = (3 + (-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 2.
%F A041007 Some properties:
%F A041007 (1) a(n)^2 - a(n-2)^2 = (3+(-1)^n)*a(2*n-1), for n > 1;
%F A041007 (2) a(2*n+1) = a(n)*(a(n+1) + a(n-1)), for n > 0;
%F A041007 (3) a(2*n) = A142239(2*n), for n >= 0;
%F A041007 (4) a(2*n+1) = A041007(2*n+1)/2, for n >= 0;
%F A041007 (5) a(2*n-1)*A142239(2*n+1) = a(n)^2 - 1, for n > 0;
%F A041007 (6) a(2*n) = a(n)*A142239(n) + a(n-1)*A142239(n-1), for n > 0;
%F A041007 (7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0..n} a(3+4*k);
%F A041007 (8) Sum_{k=0..n} (a(2*k-1) + a(2*k+1))*A142239(2*k) = Sum_{k=0..n} A142239(3+4*k). (End)
%F A041007 a(n) = ((2 + sqrt(6))^(n+1) - (2 - sqrt(6))^(n+1))/(sqrt(6) * 2^(ceiling(n/2) + 1)). - _Robert FERREOL_, Oct 14 2024
%t A041007 Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 16 2011 *)
%Y A041007 Cf. A010464, A041006.
%Y A041007 Cf. A072256, A004189.
%K A041007 nonn,cofr,frac,easy
%O A041007 0,2
%A A041007 _N. J. A. Sloane_