A041069 -id:A041069 - cp's OEIS frontend

A041068 Numerators of continued fraction convergents to sqrt(41).

6, 13, 32, 397, 826, 2049, 25414, 52877, 131168, 1626893, 3384954, 8396801, 104146566, 216689933, 537526432, 6667007117, 13871540666, 34410088449, 426792602054, 887995292557, 2202783187168, 27321393538573

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,0,64,0,0,1).

Cf. A041069, A010495.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[41],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Numerator[Convergents[Sqrt[41], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,0,64,0,0,1},{6,13,32,397,826,2049},30] (* Harvey P. Dale, Mar 17 2019 *)

G.f.: -(x^5-6*x^4+13*x^3+32*x^2+13*x+6) / (x^6+64*x^3-1). - Colin Barker, Nov 04 2013

A010133 Continued fraction for sqrt(41).

Original entry on oeis.org

6, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12, 2, 2, 12

Offset: 0

N. J. A. Sloane

			6.40312423743284868648821767... = 6 + 1/(2 + 1/(2 + 1/(12 + 1/(2 + ...)))). - _Harry J. Smith_, Jun 05 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
Index entries for continued fractions for constants
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A010495 (decimal expansion).

Cf. A041068/A041069 (convergents), A248267 (Egyptian fraction).

ContinuedFraction[Sqrt[41],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)

{ allocatemem(932245000); default(realprecision, 25000); x=contfrac(sqrt(41)); for (n=0, 20000, write("b010133.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

From Elmo R. Oliveira, Aug 03 2024: (Start)
G.f.: 2*(3 + x + x^2 + 3*x^3)/((1 - x)*(1 + x + x^2)).
a(n) = a(n-3), n > 3. (End)

cp's OEIS Frontend

A041068 Numerators of continued fraction convergents to sqrt(41).

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