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

1, 2, 9, 20, 89, 198, 881, 1960, 8721, 19402, 86329, 192060, 854569, 1901198, 8459361, 18819920, 83739041, 186298002, 828931049, 1844160100, 8205571449, 18255302998, 81226783441, 180708869880

Offset

0,2

Comments

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

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

Interspersion of 2 sequences, A072256 and 2*A004189. - Gerry Martens, Jun 10 2015

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Formula

Empirical g.f.: (1+2*x-x^2)/(1-10*x^2+x^4). - Colin Barker, Dec 31 2011

From Rogério Serôdio, Apr 01 2018: (Start)

Recurrence formula: a(n) = (3 + (-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 2.

Some properties:

(1) a(n)^2 - a(n-2)^2 = (3+(-1)^n)*a(2*n-1), for n > 1;

(2) a(2*n+1) = a(n)*(a(n+1) + a(n-1)), for n > 0;

(3) a(2*n) = A142239(2*n), for n >= 0;

(4) a(2*n+1) = A041007(2*n+1)/2, for n >= 0;

(5) a(2*n-1)*A142239(2*n+1) = a(n)^2 - 1, for n > 0;

(6) a(2*n) = a(n)*A142239(n) + a(n-1)*A142239(n-1), for n > 0;

(7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0..n} a(3+4*k);

(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)

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[6], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

Crossrefs

Cf. A010464, A041006.

Cf. A072256, A004189.

Sequence in context: A093835 A264294 A280295 * A002360 A100516 A337015

Adjacent sequences: A041004 A041005 A041006 * A041008 A041009 A041010

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved