A010495 -id:A010495 - cp's OEIS frontend

A248267 Egyptian fraction representation of sqrt(41) (A010495) using a greedy function.

6, 3, 15, 321, 111450, 533909816159, 325998701518914099105001, 1006914879088411198399682064005635831534437484321, 1497711655729721286088828059704410216184274677681054392262396421340070136379357931802690267613686

Offset: 0

Robert G. Wilson v, Oct 04 2014

nonn

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.

Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 41]]

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

Original entry on oeis.org

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

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

Original entry on oeis.org

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)

A011035 Decimal expansion of 4th root of 41.

Original entry on oeis.org

2, 5, 3, 0, 4, 3, 9, 5, 3, 4, 4, 3, 5, 2, 4, 2, 8, 7, 0, 0, 0, 3, 8, 6, 0, 2, 3, 0, 3, 0, 1, 9, 1, 4, 0, 4, 0, 7, 3, 4, 6, 8, 6, 2, 8, 5, 5, 4, 8, 2, 3, 0, 6, 7, 6, 5, 1, 6, 9, 0, 5, 6, 8, 6, 4, 0, 4, 9, 1, 2, 5, 2, 8, 2, 3, 2, 5, 3, 2, 4, 1, 3, 4, 5, 1, 6, 4, 7, 6, 3, 9, 6, 6, 7, 4, 7, 0, 2, 5

Offset: 1

N. J. A. Sloane

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Cf. A010495.

RealDigits[41^(1/4), 10, 100][[1]] (* Alonso del Arte, Sep 04 2015 *)

sqrtn(41,4) \\ Charles R Greathouse IV, Apr 15 2014

cp's OEIS Frontend

A248267 Egyptian fraction representation of sqrt(41) (A010495) using a greedy function.

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

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

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A010133 Continued fraction for sqrt(41).

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A011035 Decimal expansion of 4th root of 41.

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