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

1, 3, 19, 60, 379, 1197, 7561, 23880, 150841, 476403, 3009259, 9504180, 60034339, 189607197, 1197677521, 3782639760, 23893516081, 75463188003, 476672644099, 1505481120300, 9509559365899, 30034159217997

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: A012863 A269156 A370434 * A378115 A185448 A114250

Adjacent sequences: A041012 A041013 A041014 * A041016 A041017 A041018

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane

Status

approved