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

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

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).

Crossrefs

Cf. A010468, A041014.

Programs

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 *)

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)