A041031 Denominators of continued fraction convergents to sqrt(20).

1, 2, 17, 36, 305, 646, 5473, 11592, 98209, 208010, 1762289, 3732588, 31622993, 66978574, 567451585, 1201881744, 10182505537, 21566892818, 182717648081, 387002188980, 3278735159921, 6944472508822

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,18,0,-1).

Crossrefs

Cf. A010476, A040015, A041030 (numerators).

Programs

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[20],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
a0[n_] := ((5+2*Sqrt[5])/(9+4*Sqrt[5])^n+(5-2*Sqrt[5])*(9+4*Sqrt[5])^n)/10 //Simplify
a1[n_] := (-1/(9+4*Sqrt[5])^n+(9+4*Sqrt[5])^n)/(4*Sqrt[5]) //Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

Formula

G.f.: (1+2*x-x^2)/(1-18*x^2+x^4). - Colin Barker, Jan 01 2012

From Gerry Martens, Jul 11 2015: (Start)

Interspersion of 2 sequences [a0(n),a1(n)] for n>0:

a0(n) = ((5+2*sqrt(5))/(9+4*sqrt(5))^n+(5-2*sqrt(5))*(9+4*sqrt(5))^n)/10.

a1(n) = (-1/(9+4*sqrt(5))^n+(9+4*sqrt(5))^n)/(4*sqrt(5)). (End)