A041117 Denominators of continued fraction convergents to sqrt(67).

1, 5, 11, 16, 27, 205, 232, 437, 1106, 5967, 96578, 488857, 1074292, 1563149, 2637441, 20025236, 22662677, 42687913, 108038503, 582880428, 9434125351, 47753507183, 104941139717, 152694646900, 257635786617, 1956145153219, 2213780939836, 4169926093055

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 (0,0,0,0,0,0,0,0,0,97684,0,0,0,0,0,0,0,0,0,-1).

Crossrefs

Cf. A041116, A010147, A020824, A010519.

Programs

Magma

I:=[1, 5, 11, 16, 27, 205, 232, 437, 1106, 5967, 96578, 488857, 1074292, 1563149, 2637441, 20025236, 22662677, 42687913, 108038503, 582880428]; [n le 20 select I[n] else 97684*Self(n-10)-Self(n-20): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Mathematica

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[67],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Denominator[Convergents[Sqrt[67], 30]] (* Harvey P. Dale, Oct 03 2012 *)
CoefficientList[Series[-(x^18 - 5 x^17 + 11 x^16 - 16 x^15 + 27 x^14 - 205 x^13 + 232 x^12 - 437 x^11 + 1106 x^10 - 5967 x^9 - 1106 x^8 - 437 x^7 - 232 x^6 - 205 x^5 - 27 x^4 - 16 x^3 - 11 x^2 - 5 x - 1)/(x^20 - 97684 x^10 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

Formula

G.f.: -(x^18 -5*x^17 +11*x^16 -16*x^15 +27*x^14 -205*x^13 +232*x^12 -437*x^11 +1106*x^10 -5967*x^9 -1106*x^8 -437*x^7 -232*x^6 -205*x^5 -27*x^4 -16*x^3 -11*x^2 -5*x -1) / (x^20 -97684*x^10 +1). - Colin Barker, Nov 13 2013

a(n) = 97684*a(n-10) - a(n-20). - Vincenzo Librandi, Dec 11 2013

Extensions

More terms from Colin Barker, Nov 13 2013