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A041015 Denominators of continued fraction convergents to sqrt(11). 7
1, 3, 19, 60, 379, 1197, 7561, 23880, 150841, 476403, 3009259, 9504180, 60034339, 189607197, 1197677521, 3782639760, 23893516081, 75463188003, 476672644099, 1505481120300, 9509559365899, 30034159217997 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

FORMULA

G.f.: (1+3*x-x^2)/(1-20*x^2+x^4). - Colin Barker, Dec 31 2011

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)]:

a0(n) = ((11+3*sqrt(11))/(10+3*sqrt(11))^n+(11-3*sqrt(11))*(10+3*sqrt(11))^n)/22.

a1(n) = 3*sum(i=1,n,a0(i)). (End)

MATHEMATICA

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[11], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

a0[n_] := (11+3*Sqrt[11]+(11-3*Sqrt[11])*(10+3*Sqrt[11])^(2*n))/(22*(10+3*Sqrt[11])^n) // Simplify

a1[n_] := 3*Sum[a0[i], {i, 1, n}]

Flatten[MapIndexed[{a0[#], a1[#]}&, Range[11]]] (* Gerry Martens, Jul 10 2015 *)

CROSSREFS

Cf. A010468, A041014.

Sequence in context: A182353 A012863 A269156 * A185448 A114250 A249994

Adjacent sequences:  A041012 A041013 A041014 * A041016 A041017 A041018

KEYWORD

nonn,cofr,frac,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 28 21:37 EDT 2020. Contains 334690 sequences. (Running on oeis4.)