A041068 -id:A041068 - cp's OEIS frontend

A041069 Denominators of continued fraction convergents to sqrt(41).

1, 2, 5, 62, 129, 320, 3969, 8258, 20485, 254078, 528641, 1311360, 16264961, 33841282, 83947525, 1041211582, 2166370689, 5373952960, 66653806209, 138681565378, 344016936965, 4266884808958, 8877786554881, 22022457918720, 273147281579521, 568317021077762

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,64,0,0,1).

Cf. A010495, A041068.

I:=[1, 2, 5, 62, 129, 320]; [n le 6 select I[n] else 64*Self(n-3)+Self(n-6): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[41], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Denominator[Convergents[Sqrt[41], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{0,0,64,0,0,1},{1,2,5,62,129,320},40] (* Harvey P. Dale, Jun 19 2022 *)

G.f.: -(x^4-2*x^3+5*x^2+2*x+1) / (x^6+64*x^3-1). - Colin Barker, Nov 12 2013

a(n) = 64*a(n-3) + a(n-6). - Vincenzo Librandi, Dec 10 2013

More terms from Colin Barker, Nov 12 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

A041069 Denominators of continued fraction convergents to sqrt(41).

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