A010481 -id:A010481 - cp's OEIS frontend

A248253 Egyptian fraction representation of sqrt(26) (A010481) using a greedy function.

5, 11, 124, 21784, 767400293, 1762025132544871871, 3756028786746097256770667892973677974, 42736560346010944990137576929510502074095427615068285034007804816583306199

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[ 26]]

A040020 Continued fraction for sqrt(26).

Original entry on oeis.org

5, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 0

N. J. A. Sloane

			5.09901951359278483002822... = 5 + 1/(10 + 1/(10 + 1/(10 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 03 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 (1).

Cf. A010481 (decimal expansion), A041040/A041041 (convergents), A248253 (Egyptian fraction).

Cf. A040000, A010692.

numtheory[cfrac](sqrt(26), 100, 'quotients'); # obsolete code updated by Alois P. Heinz, Feb 24 2018

ContinuedFraction[Sqrt[26],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{5},120,{10}] (* Harvey P. Dale, Apr 22 2021 *)

contfrac(sqrt(26)) \\ Harry J. Smith, Jun 03 2009 [Edited by M. F. Hasler, Feb 24 2018]

A040020(n)=if(n,10,5) \\ M. F. Hasler, Feb 24 2018

From Elmo R. Oliveira, Feb 06 2024: (Start)
a(n) = 10 for n >= 1.
G.f.: 5*(1+x)/(1-x).
E.g.f.: 10*exp(x) - 5.
a(n) = 5*A040000(n). (End)

A176537 Decimal expansion of 5 + sqrt(26).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 24 2010

Continued fraction expansion of 5 + sqrt(26) is A010692.

This is the shape of a 10-extension rectangle; see A188640 for definitions. - Clark Kimberling, Apr 09 2011

			5+sqrt(26) = 10.09901951359278483002...

Daniel Starodubtsev, Table of n, a(n) for n = 2..10000
Wikipedia, Metallic mean
Index entries for algebraic numbers, degree 2.

Cf. A010481 (decimal expansion of sqrt(26)), A010692 (all 10's sequence).

r=10; t = (r + (4+r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]]
RealDigits[5+Sqrt[26],10,120][[1]] (* Harvey P. Dale, Jun 24 2013 *)

5+sqrt(26) \\ Michel Marcus, Jul 23 2018

a(n) = A010481(n-2) for n > 3.
Equals exp(arcsinh(5)), since arcsinh(x) = log(x + sqrt(x^2 + 1)). - Stanislav Sykora, Nov 01 2013
Equals limit_{n->infinity} S(n, 2*sqrt(2*13))/ S(n-1, 2*sqrt(2*13)), with the S-Chebyshev polynomilas (see A049310). - Wolfdieter Lang, Nov 15 2023

A188659 Decimal expansion of (1+sqrt(26))/5.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, Apr 09 2011

Decimal expansion of the shape of a (2/5)-extension rectangle; see A188640 for definitions of shape and r-extension rectangle. Briefly, shape=length/width, and an r-extension rectangle is composed of two rectangles of shape 1/r when r<1.

The continued fraction of the constant is 1, 4, 1, 1, 4, 1, ... = A146325.

			1.219803902718556966005644821804556397912754189219919281516994...

Index entries for algebraic numbers, degree 2.

Cf. A146325, A188640, A188658, A188729, A188730, A010481.

RealDigits[(1 + Sqrt[26])/5, 10, 111][[1]] (* Robert G. Wilson v, Aug 18 2011 *)

Equals exp(arcsinh(1/5)). - Amiram Eldar, Jul 04 2023

A041040 Numerators of continued fraction convergents to sqrt(26).

Original entry on oeis.org

5, 51, 515, 5201, 52525, 530451, 5357035, 54100801, 546365045, 5517751251, 55723877555, 562756526801, 5683289145565, 57395647982451, 579639768970075, 5853793337683201, 59117573145802085, 597029524795704051

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (10,1).

Cf. A010481, A041041.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[26],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
CoefficientList[Series[(5 + x)/(1 - 10 x - x^2), {x, 0, 30}], x]  (* Vincenzo Librandi_, Oct 28 2013 *)

From Philippe Deléham, Nov 20 2008: (Start)
a(n) = 10*a(n-1) + a(n-2), a(0)=5, a(1)=51.
G.f.: (5+x)/(1-10*x-x^2). (End)

A177153 Decimal expansion of (21+5*sqrt(26))/19.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 03 2010

Continued fraction expansion of (21+5*sqrt(26))/19 is A165207.

			(21+5*sqrt(26))/19 = 2.44711039831389074474...

Cf. A010481 (decimal expansion of sqrt(26)), A165207 (repeat 2, 2, 4, 4).

RealDigits[(21+5Sqrt[26])/19,10,120][[1]] (* Harvey P. Dale, Jul 26 2020 *)

A373053 Decimal expansion of sqrt(26)/2.

Original entry on oeis.org

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

Offset: 1

Gonzalo Martínez, May 20 2024

sqrt(26)/2 is the area of a triangle of sides sqrt(5), sqrt(6) and sqrt(7), which is the smallest triangle whose sides are of the form sqrt(k), sqrt(k + 1) and sqrt(k + 2) and are all irrational numbers, where k is a positive integer.

			2.54950975679639241501411...

Cf. A010481.

Mathematica
```
RealDigits[Sqrt[26]/2, 10, 200][[1]]
```

Equals sqrt(13/2).

cp's OEIS Frontend

A248253 Egyptian fraction representation of sqrt(26) (A010481) using a greedy function.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A040020 Continued fraction for sqrt(26).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A176537 Decimal expansion of 5 + sqrt(26).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A188659 Decimal expansion of (1+sqrt(26))/5.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A041040 Numerators of continued fraction convergents to sqrt(26).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A177153 Decimal expansion of (21+5*sqrt(26))/19.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A373053 Decimal expansion of sqrt(26)/2.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula