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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A010494, A041066 (numerators).

Sequence in context: A300636 A301358 A163156 * A046867 A154823 A109835

Adjacent sequences: A041064 A041065 A041066 * A041068 A041069 A041070

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane.

Extensions

More terms from Colin Barker, Nov 12 2013

Status

approved