A041029 -id:A041029 - cp's OEIS frontend

A010124 Continued fraction for sqrt(19).

4, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1, 3, 1, 2, 8, 2, 1

Offset: 0

N. J. A. Sloane

			4.358898943540673552236981983... = 4 + 1/(2 + 1/(1 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 03 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac
A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A041028/A041029 (convergents).

Cf. A010475 (decimal expansion).

ContinuedFraction[Sqrt[19],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)

{ allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(19)); for (n=0, 20000, write("b010124.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

G.f.: (4 + 2*x + x^2 + 3*x^3 + x^4 + 2*x^5 + 4*x^6)/(1 - x^6). - Stefano Spezia, Jul 26 2025

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

Original entry on oeis.org

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

N. J. A. Sloane

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).

Cf. A010124, A010475, A041029.

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 *)

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]

cp's OEIS Frontend

A010124 Continued fraction for sqrt(19).

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