A010496 -id:A010496 - cp's OEIS frontend

A248268 Egyptian fraction representation of sqrt(42) (A010496) using a greedy function.

6, 3, 7, 220, 209746, 1800026104632, 11289682294671072755879655, 1247832270676194041105480584245717817404868332358363, 5623554373314472317858205865619051220489843727752125404940182021329874216730979924009375686764591034334

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[ 42]]

A040035 Continued fraction for sqrt(42).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			6.4807406984078602309659674... = 6 + 1/(2 + 1/(12 + 1/(2 + 1/(12 + ...)))). - _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, 1).

Cf. A010496 (decimal expansion), A040011.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[42],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)

{ allocatemem(932245000); default(realprecision, 29000); x=contfrac(sqrt(42)); for (n=0, 20000, write("b040035.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 2*A040011(n).
G.f.: 2*(3 + x + 3*x^2)/(1 - x^2). (End)

A041070 Numerators of continued fraction convergents to sqrt(42).

Original entry on oeis.org

6, 13, 162, 337, 4206, 8749, 109194, 227137, 2834838, 5896813, 73596594, 153090001, 1910676606, 3974443213, 49603995162, 103182433537, 1287793197606, 2678768828749, 33433019142594, 69544807113937

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 (0,26,0,-1).

Cf. A041071, A010496.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[42],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011*)
Numerator[Convergents[Sqrt[42], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
LinearRecurrence[{0,26,0,-1},{6,13,162,337},30] (* Harvey P. Dale, Aug 07 2019 *)

G.f.: -(x+1)*(x^2-7*x-6) / (x^4-26*x^2+1). - Colin Barker, Nov 04 2013

A041071 Denominators of continued fraction convergents to sqrt(42).

Original entry on oeis.org

1, 2, 25, 52, 649, 1350, 16849, 35048, 437425, 909898, 11356201, 23622300, 294823801, 613269902, 7654062625, 15921395152, 198710804449, 413343004050, 5158826853049, 10730996710148, 133930787374825, 278592571459798, 3477041644892401, 7232675861244600

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 (0,26,0,-1).

Cf. A010496, A040035, A041070 (numerators).

I:=[1, 2, 25, 52]; [n le 4 select I[n] else 26*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[42],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 22 2011 *)
Denominator[Convergents[Sqrt[42], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
LinearRecurrence[{0,26,0,-1},{1,2,25,52},30] (* Harvey P. Dale, Oct 17 2019 *)

G.f.: -(x^2-2*x-1) / (x^4-26*x^2+1). - Colin Barker, Nov 12 2013

a(n) = 26*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

More terms from Colin Barker, Nov 12 2013

A176107 Decimal expansion of (6+sqrt(42))/4.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 10 2010

Continued fraction expansion of (6+sqrt(42))/4 is A010706.

			(6+sqrt(42))/4 = 3.12018517460196505774...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010496 (decimal expansion of sqrt(42)), A010706 (repeat 3, 8).

RealDigits[(6+Sqrt[42])/4,10,120][[1]] (* Harvey P. Dale, Jun 22 2012 *)

A176216 Decimal expansion of (6+sqrt(42))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 12 2010

Continued fraction expansion of (6+sqrt(42))/3 is A010711.

			(6+sqrt(42))/3 = 4.16024689946928674365...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010496 (decimal expansion of sqrt(42)), A010711 (repeat 4, 6).

A176396 Decimal expansion of (6+sqrt(42))/2.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 16 2010

Continued fraction expansion of (6+sqrt(42))/2 is A010711 preceded by 6.

			(6+sqrt(42))/2 = 6.24037034920393011548...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010496 (decimal expansion of sqrt(42)), A010711 (repeat 4, 6).

RealDigits[(6+Sqrt[42])/2,10,120][[1]] (* Harvey P. Dale, May 25 2012 *)

A176454 Decimal expansion of (12+2*sqrt(42))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of (12+2*sqrt(42))/3 is A010706 preceded by 8.

			(12+2*sqrt(42))/3 = 8.32049379893857348731...

Cf. A010496 (decimal expansion of sqrt(42)), A010706 (repeat 3, 8).

RealDigits[(12+2*Sqrt[42])/3,10,120][[1]] (* Harvey P. Dale, Nov 23 2016 *)

cp's OEIS Frontend

A248268 Egyptian fraction representation of sqrt(42) (A010496) using a greedy function.

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A040035 Continued fraction for sqrt(42).

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A041070 Numerators of continued fraction convergents to sqrt(42).

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A041071 Denominators of continued fraction convergents to sqrt(42).

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A176107 Decimal expansion of (6+sqrt(42))/4.

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A176216 Decimal expansion of (6+sqrt(42))/3.

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A176396 Decimal expansion of (6+sqrt(42))/2.

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A176454 Decimal expansion of (12+2*sqrt(42))/3.

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Mathematica