A010492 -id:A010492 - cp's OEIS frontend

A248264 Egyptian fraction representation of sqrt(38) (A010492) using a greedy function.

6, 7, 47, 3569, 13543237, 813461964457561, 7421316108781190769825230152615, 711253293828537228004750977021512448161146012227144474046636992, 2200029703970808428058199608953702518884689809814432014002394662129432102727790523039076189301028040002865113400234535183784056

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[ 38]]

A040031 Continued fraction for sqrt(38).

Original entry on oeis.org

6, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6, 12, 6

Offset: 0

N. J. A. Sloane

			6.1644140029689764502501923... = 6 + 1/(6 + 1/(12 + 1/(6 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 04 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. A010492 (decimal expansion), A040001.

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

ContinuedFraction[Sqrt[38],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)

{ allocatemem(932245000); default(realprecision, 38000); x=contfrac(sqrt(38)); for (n=0, 20000, write("b040031.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 6*A040001(n).
G.f.: 6*(1 + x + x^2)/(1 - x^2). (End)

A041062 Numerators of continued fraction convergents to sqrt(38).

Original entry on oeis.org

6, 37, 450, 2737, 33294, 202501, 2463306, 14982337, 182251350, 1108490437, 13484136594, 82013310001, 997643856606, 6067876449637, 73812161252250, 448940843963137, 5461102288809894, 33215554576822501

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 (0,74,0,-1).

Cf. A041063, A010492.

Numerator[Convergents[Sqrt[38], 30]] (* Vincenzo Librandi, Oct 29 2013 *)
a0[n_] := (-3+Sqrt[19/2])*(37+6*Sqrt[38])^n-(6+Sqrt[38])/(2*(37+6*Sqrt[38])^n) // Simplify
a1[n_] := (1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/2 // FullSimplify
Flatten[MapIndexed[{a0[#], a1[#]}&, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,74,0,-1},{6,37,450,2737},20] (* Harvey P. Dale, Oct 17 2020 *)

G.f.: -(x^3-6*x^2-37*x-6) / (x^4-74*x^2+1). - Colin Barker, Nov 04 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = (-3+sqrt(19/2))*(37+6*sqrt(38))^n-(6+sqrt(38))/(2*(37+6*sqrt(38))^n).
a1(n) = (1/(37+6*sqrt(38))^n+(37+6*sqrt(38))^n)/2. (End)

A041063 Denominators of continued fraction convergents to sqrt(38).

Original entry on oeis.org

1, 6, 73, 444, 5401, 32850, 399601, 2430456, 29565073, 179820894, 2187415801, 13304315700, 161839204201, 984339540906, 11973913695073, 72827821711344, 885907774231201, 5388274467098550, 65545201379413801, 398659482743581356, 4849458994302390073

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 (0,74,0,-1).

Cf. A010492, A041062.

I:=[1, 6, 73, 444]; [n le 4 select I[n] else 74*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[38], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011*)
Denominator[Convergents[Sqrt[38], 30]] (* Vincenzo Librandi, Dec 10 2013 *)
a0[n_] := ((38+6*Sqrt[38])/(37+6*Sqrt[38])^n+(38-6*Sqrt[38])*(37+6*Sqrt[38])^n)/76 // Simplify
a1[n_] := (-1/(37+6*Sqrt[38])^n+(37+6*Sqrt[38])^n)/(2*Sqrt[38]) // FullSimplify
Flatten[MapIndexed[{a0[#],a1[#]}&, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
LinearRecurrence[{0,74,0,-1},{1,6,73,444},30] (* Harvey P. Dale, Feb 29 2024 *)

G.f.: -(x^2-6*x-1) / (x^4-74*x^2+1). - Colin Barker, Nov 12 2013
a(n) = 74*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = ((38+6*sqrt(38))/(37+6*sqrt(38))^n+(38-6*sqrt(38))*(37+6*sqrt(38))^n)/76;
a1(n) = (-1/(37+6*sqrt(38))^n+(37+6*sqrt(38))^n)/(2*sqrt(38)). (End)

More terms from Colin Barker, Nov 12 2013

A176459 Decimal expansion of (12+2*sqrt(38))/3.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 20 2010

Continued fraction expansion of (12+2*sqrt(38))/3 is A010732.

			(12+2*sqrt(38))/3 = 8.10960933531265096683...

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

Cf. A010492 (decimal expansion of sqrt(38)), A010732 (repeat 8, 9).

RealDigits[(12+2*Sqrt[38])/3,10,120][[1]] (* Harvey P. Dale, Aug 10 2020 *)

sqrt(152)/3+4 \\ Charles R Greathouse IV, Apr 07 2020

A176521 Decimal expansion of (18+3*sqrt(38))/4.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 23 2010

Continued fraction expansion of (18+3*sqrt(38))/4 is A010732 preceded by 9.

			(18+3*sqrt(38))/4 = 9.12331050222673233768...

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

Cf. A010492 (decimal expansion of sqrt(38)), A010732 (repeat 8, 9).

RealDigits[(3*Sqrt[38] + 18)/4, 10, 100][[1]] (* Amiram Eldar, May 01 2021 *)

(3*sqrt(38)+18)/4 \\ Charles R Greathouse IV, Apr 25 2016

cp's OEIS Frontend

A248264 Egyptian fraction representation of sqrt(38) (A010492) using a greedy function.

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A040031 Continued fraction for sqrt(38).

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A041062 Numerators of continued fraction convergents to sqrt(38).

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A041063 Denominators of continued fraction convergents to sqrt(38).

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A176459 Decimal expansion of (12+2*sqrt(38))/3.

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A176521 Decimal expansion of (18+3*sqrt(38))/4.

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