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

0,1

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).

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]

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 *)

Crossrefs

Cf. A010124, A010475, A041029.

Sequence in context: A033287 A041323 A319217 * A041211 A042323 A042083

Adjacent sequences: A041025 A041026 A041027 * A041029 A041030 A041031

Keyword

nonn,cofr,frac,easy

Author

N. J. A. Sloane.

Status

approved