A041028 -id:A041028 - 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A. J. van der Poorten, An introduction to continued fractions, Unpublished.
A. J. van der Poorten, An introduction to continued fractions, Unpublished [Cached copy]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

A041029 Denominators of continued fraction convergents to sqrt(19).

Original entry on oeis.org

1, 2, 3, 11, 14, 39, 326, 691, 1017, 3742, 4759, 13260, 110839, 234938, 345777, 1272269, 1618046, 4508361, 37684934, 79878229, 117563163, 432567718, 550130881, 1532829480, 12812766721, 27158362922

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, A041028.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[19],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
CoefficientList[Series[- (x^10 - 2 x^9 + 3 x^8 - 11 x^7 + 14 x^6 - 39 x^5 - 14 x^4 - 11 x^3 - 3 x^2 - 2 x - 1)/(x^12 - 340 x^6 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)

a(n) = 340*a(n-6)-a(n-12). G.f.: -(x^10- 2*x^9+ 3*x^8 -11*x^7 +14*x^6 -39*x^5 -14*x^4 -11*x^3 -3*x^2 -2*x -1)/(x^12-340*x^6+1). [Colin Barker, Jul 16 2012]

A341862 a(n) is the even term in the linear recurrence signature for numerators and denominators of continued fraction convergents to sqrt(n), or 0 if n is a square.

Original entry on oeis.org

0, 0, 2, 4, 0, 4, 10, 16, 6, 0, 6, 20, 14, 36, 30, 8, 0, 8, 34, 340, 18, 110, 394, 48, 10, 0, 10, 52, 254, 140, 22, 3040, 34, 46, 70, 12, 0, 12, 74, 50, 38, 64, 26, 6964, 398, 322, 48670, 96, 14, 0, 14, 100, 1298, 364, 970, 178, 30, 302, 198, 1060, 62, 59436

Offset: 0

Georg Fischer, Feb 22 2021

nonn

The Everest et al. link states that "the continued fraction expansion of a quadratic irrational is eventually periodic, which implies that the numerators px and denominators qx of its convergents satisfy linear recurrence relations".
Let k be the period length minus one of the continued fraction of sqrt(n). Then the linear recurrence signatures with constant coefficients have the form (0, 0, ..., 0, a(n), 0, 0, ..., 0, (-1)^(n+1)), with k zeroes before and behind a(n).
a(n) is twice the numerator of the convergent to sqrt(n) with index k (starting with 0).
These properties result from the mirrored structure of the period of such continued fractions.
The sequence has remarkably many terms in common with A180495 and with 2*A033313.

			The numerators for sqrt(13) begin with 3, 4, 7, 11, 18, 119, ... (A041018) and have the signature (0,0,0,0,36,0,0,0,0,1). The continued fraction has period [1,1,1,1,6], so k=4 and a(13) = 2*A041018(4) = 2*18 = 36. The signature ends with (-1)^4.
The numerators for sqrt(19) begin with 4, 9, 13, 48, 61, 170, 1421, ... (A041028) and have the signature (0,0,0,0,0,340,0,0,0,0,0,-1). The continued fraction has period [2,1,3,1,2,8], so k=5 and a(19) = 2*A041028(5) = 2*170 = 340. The signature ends with (-1)^5.

Jean-François Alcover, Table of n, a(n) for n = 0..10000
Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence Sequences, AMS Mathematical Surveys and Monographs, Volume 104 (2003) p. 8, 5th paragraph.

Cf. A006702, A033313, A180495.

a(n) = 2*A006702(n) if n is not square, otherwise 0.

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A010124 Continued fraction for sqrt(19).

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A041029 Denominators of continued fraction convergents to sqrt(19).

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A341862 a(n) is the even term in the linear recurrence signature for numerators and denominators of continued fraction convergents to sqrt(n), or 0 if n is a square.

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