A010501 -id:A010501 - cp's OEIS frontend

A248273 Egyptian fraction representation of sqrt(47) (A010501) using a greedy function.

6, 2, 3, 45, 10097, 180933939, 70214804893433857, 24596197522004292913199742834240369, 851917396155337556711167562167009352482986581505723891411145951010937, 1830843559366860042528367793031819716270540620095563249767306742965459078226069734667092696644523226923832775331940549734586475295256730688

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[ 47]]

A010137 Continued fraction for sqrt(47).

Original entry on oeis.org

6, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12, 1, 5, 1, 12

Offset: 0

N. J. A. Sloane

			6.85565460040104412493587144... = 6 + 1/(1 + 1/(5 + 1/(1 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 06 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 (0,0,0,1).

Cf. A010501 (decimal expansion).

ContinuedFraction[Sqrt[47],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(47)); for (n=0, 20000, write("b010137.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 06 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 5, a(2^e) = 12 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(s-2) + 7/4^s). (End)
G.f.: (6 + x + 5*x^2 + x^3 + 6*x^4)/(1 - x^4). - Stefano Spezia, Jul 27 2025

A176524 Decimal expansion of sqrt(235).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of sqrt(235) is A040219.

			sqrt(235) = 15.32970971675589165655...

Cf. A002163 (decimal expansion of sqrt(5)), A010501 (decimal expansion of sqrt(47)), A176523 (decimal expansion of (45+3*sqrt(235))/10), A040219 (15 followed by (repeat 3, 30)).

RealDigits[Sqrt[235],10,120][[1]] (* Harvey P. Dale, Mar 12 2015 *)

A041080 Numerators of continued fraction convergents to sqrt(47).

Original entry on oeis.org

6, 7, 41, 48, 617, 665, 3942, 4607, 59226, 63833, 378391, 442224, 5685079, 6127303, 36321594, 42448897, 545708358, 588157255, 3486494633, 4074651888, 52382317289, 56456969177, 334667163174, 391124132351, 5028156751386, 5419280883737, 32124561170071

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 (0,0,0,96,0,0,0,-1).

Cf. A041081, A010501.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[47],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Numerator[Convergents[Sqrt[47],30]] (* Harvey P. Dale, Oct 15 2013 *)

G.f.: -(x^7-6*x^6+7*x^5-41*x^4-48*x^3-41*x^2-7*x-6) / (x^8-96*x^4+1). - Colin Barker, Nov 04 2013

More terms from Colin Barker, Nov 04 2013

A041081 Denominators of continued fraction convergents to sqrt(47).

Original entry on oeis.org

1, 1, 6, 7, 90, 97, 575, 672, 8639, 9311, 55194, 64505, 829254, 893759, 5298049, 6191808, 79599745, 85791553, 508557510, 594349063, 7640746266, 8235095329, 48816222911, 57051318240, 733432041791, 790483360031, 4685848841946, 5476332201977, 70401835265670

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 (0,0,0,96,0,0,0,-1).

Cf. A010501, A041080.

I:=[1, 1, 6, 7, 90, 97, 575, 672]; [n le 8 select I[n] else 96*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[47],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011 *)
CoefficientList[Series[(1 + x + 6 x^2 + 7 x^3 - 6 x^4 + x^5 - x^6)/(x^8 - 96 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)

G.f.: -(x^2-x-1)*(x^4+7*x^2+1) / (x^8-96*x^4+1). - Colin Barker, Nov 12 2013

a(n) = 96*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013

More terms from Colin Barker, Nov 12 2013

cp's OEIS Frontend

A248273 Egyptian fraction representation of sqrt(47) (A010501) using a greedy function.

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A010137 Continued fraction for sqrt(47).

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A176524 Decimal expansion of sqrt(235).

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A041080 Numerators of continued fraction convergents to sqrt(47).

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A041081 Denominators of continued fraction convergents to sqrt(47).

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