A010522 -id:A010522 - cp's OEIS frontend

A248294 Egyptian fraction representation of sqrt(70) (A010522) using a greedy function.

8, 3, 31, 992, 1245369, 3302336350417, 47523810173595463077699706, 15227181289661678179456803859437449044352739723867580, 290150350103448613285887398334236111049315440797539935407545942151460489216853681370927408862165807652692

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[ 70]]

A010149 Continued fraction for sqrt(70).

Original entry on oeis.org

8, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1, 2, 1, 2, 16, 2, 1

Offset: 0

N. J. A. Sloane

			8.366600265340755479781720257... = 8 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + ...)))). - _Harry J. Smith_, Jun 08 2009

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, 0, 0, 1).

Cf. A010522 Decimal expansion. - Harry J. Smith, Jun 08 2009

ContinuedFraction[Sqrt[70],300] (* Vladimir Joseph Stephan Orlovsky, Mar 08 2011 *)
PadRight[{8},120,{16,2,1,2,1,2}] (* Harvey P. Dale, Mar 25 2020 *)

{ allocatemem(932245000); default(realprecision, 19000); x=contfrac(sqrt(70)); for (n=0, 20000, write("b010149.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 08 2009

A041122 Numerators of continued fraction convergents to sqrt(70).

Original entry on oeis.org

8, 17, 25, 67, 92, 251, 4108, 8467, 12575, 33617, 46192, 126001, 2062208, 4250417, 6312625, 16875667, 23188292, 63252251, 1035224308, 2133700867, 3168925175, 8471551217, 11640476392, 31752504001

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 (0,0,0,0,0,502,0,0,0,0,0,-1).

Cf. A010522, A041123.

Numerator[Convergents[Sqrt[70], 30]] (* Vincenzo Librandi, Oct 26 2013 *)
LinearRecurrence[{0,0,0,0,0,502,0,0,0,0,0,-1},{8,17,25,67,92,251,4108,8467,12575,33617,46192,126001},30] (* Harvey P. Dale, Dec 12 2024 *)

a(n) = 502*a(n-6)-a(n-12). G.f.: -(x+1)*(x^10-9*x^9+26*x^8-51*x^7+118*x^6-210*x^5-41*x^4-51*x^3-16*x^2-9*x-8)/(x^12-502*x^6+1). [Colin Barker, Jul 18 2012]

A041123 Denominators of continued fraction convergents to sqrt(70).

Original entry on oeis.org

1, 2, 3, 8, 11, 30, 491, 1012, 1503, 4018, 5521, 15060, 246481, 508022, 754503, 2017028, 2771531, 7560090, 123732971, 255026032, 378759003, 1012544038, 1391303041, 3795150120, 62113704961, 128022560042

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 (0,0,0,0,0,502,0,0,0,0,0,-1).

Cf. A010522, A041122.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[70],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Denominator[Convergents[Sqrt[70], 30]] (* Vincenzo Librandi, Oct 24 2013 *)

a(n) = 502*a(n-6)-a(n-12). G.f.: -(x^2-2*x-1)*(x^8+4*x^6+15*x^4+4*x^2+1)/ (x^12-502*x^6+1). [Colin Barker, Jul 18 2012]

A020827 Decimal expansion of 1/sqrt(70).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.11952286.. = 1/A010522.

Ivan Panchenko, Table of n, a(n) for n = 0..1000

RealDigits[1/Sqrt[70],10,120][[1]] (* Harvey P. Dale, Jul 03 2017 *)

A171542 Decimal expansion of sqrt(27/70).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Dec 11 2009

The absolute value of the Clebsch-Gordan coupling coefficient = <2 3/2 ; -1 3/2 | 5/2 1/2>.

			sqrt(27/70) = 3*sqrt(210)/70 = 0.621059003408118796016514178164...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Wikipedia, Table of Clebsch-Gordan coefficients

Cf. A010482, A010522, A010541, A171537.

RealDigits[Sqrt[27/70],10,120][[1]] (* Harvey P. Dale, Apr 26 2011 *)

Equals A010482/A010522 = A010541*A171537/10.

cp's OEIS Frontend

A248294 Egyptian fraction representation of sqrt(70) (A010522) using a greedy function.

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A010149 Continued fraction for sqrt(70).

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A041122 Numerators of continued fraction convergents to sqrt(70).

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A041123 Denominators of continued fraction convergents to sqrt(70).

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A020827 Decimal expansion of 1/sqrt(70).

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A171542 Decimal expansion of sqrt(27/70).

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