A041034 Numerators of continued fraction convergents to sqrt(22).

4, 5, 14, 61, 136, 197, 1712, 1909, 5530, 24029, 53588, 77617, 674524, 752141, 2178806, 9467365, 21113536, 30580901, 265760744, 296341645, 858444034, 3730117781, 8318679596, 12048797377, 104709058612

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,394,0,0,0,0,0,-1).

Crossrefs

Cf. A010478, A041035.

Programs

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[22],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
CoefficientList[Series[- (x^11 - 4 x^10 + 5 x^9 - 14 x^8 + 61 x^7 - 136 x^6 - 197 x^5 - 136 x^4 - 61 x^3 - 14 x^2 - 5 x - 4)/(x^12 - 394 x^6 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{0,0,0,0,0,394,0,0,0,0,0,-1},{4,5,14,61,136,197,1712,1909,5530,24029,53588,77617},30] (* Harvey P. Dale, Mar 14 2017 *)

Formula

a(n) = 394*a(n-6)-a(n-12). G.f.: -(x^11 -4*x^10 +5*x^9 -14*x^8 +61*x^7 -136*x^6 -197*x^5 -136*x^4 -61*x^3 -14*x^2 -5*x -4)/(x^12-394*x^6+1). [Colin Barker, Jul 16 2012]