A041028 Numerators of continued fraction convergents to sqrt(19).

4, 9, 13, 48, 61, 170, 1421, 3012, 4433, 16311, 20744, 57799, 483136, 1024071, 1507207, 5545692, 7052899, 19651490, 164264819, 348181128, 512445947, 1885518969, 2397964916, 6681448801, 55849555324

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 (0,0,0,0,0,340,0,0,0,0,0,-1).

Crossrefs

Cf. A010124, A010475, A041029.

Programs

Mathematica

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[19],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
CoefficientList[Series[- (x^11 - 4 x^10 + 9 x^9 - 13 x^8 + 48 x^7 - 61 x^6 - 170 x^5 - 61 x^4 - 48 x^3 - 13 x^2 - 9 x - 4)/(x^12 - 340 x^6 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{0,0,0,0,0,340,0,0,0,0,0,-1},{4,9,13,48,61,170,1421,3012,4433,16311,20744,57799},30] (* or *) Numerator[ Convergents[ Sqrt[19],30]] (* Harvey P. Dale, Jul 04 2021 *)

Formula

a(n) = 340*a(n-6)-a(n-12). G.f.: -(x^11 -4*x^10 +9*x^9 -13*x^8 +48*x^7 -61*x^6 -170*x^5 -61*x^4 -48*x^3 -13*x^2 -9*x -4)/(x^12-340*x^6+1). [Colin Barker, Jul 16 2012]