A041067 Denominators of continued fraction convergents to sqrt(40).

1, 3, 37, 114, 1405, 4329, 53353, 164388, 2026009, 6242415, 76934989, 237047382, 2921503573, 9001558101, 110940200785, 341822160456, 4212806126257, 12980240539227, 159975692596981, 492907318330170, 6074863512559021, 18717497856007233, 230684837784645817

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Comments

With a(-1) = 0, a(n-1) gives, for n >= 0, the numerator of the convergents to 1/sqrt(40) = 1/(2*sqrt(10)) = A020797. - Wolfdieter Lang, Nov 21 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (0,38,0,-1).

Crossrefs

Cf. A010494, A041066 (numerators).

Programs

Magma

I:=[1, 3, 37, 114]; [n le 4 select I[n] else 38*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[40],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Denominator[Convergents[Sqrt[40],30]] (* Harvey P. Dale, Sep 12 2013 *)
CoefficientList[Series[-(x^2 - 3 x - 1)/((x^2 - 6 x - 1)(x^2 + 6 x - 1)), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 10 2013 *)

Formula

G.f.: -(x^2-3*x-1) / ((x^2-6*x-1)*(x^2+6*x-1)). - Colin Barker, Nov 12 2013

a(n) = 38*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

Extensions

More terms from Colin Barker, Nov 12 2013