A010484 -id:A010484 - cp's OEIS frontend

A248256 Egyptian fraction representation of sqrt(29) (A010484) using a greedy function.

5, 3, 20, 547, 301184, 147558270953, 86497522148984105061516, 8551929116782420428265616715584087619312418621, 99749931990938468116836650873165146389082138700427586581720209715910550531363814159816345424

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[ 29]]

A041046 Numerators of continued fraction convergents to sqrt(29).

Original entry on oeis.org

5, 11, 16, 27, 70, 727, 1524, 2251, 3775, 9801, 101785, 213371, 315156, 528527, 1372210, 14250627, 29873464, 44124091, 73997555, 192119201, 1995189565, 4182498331, 6177687896, 10360186227, 26898060350

Offset: 0

N. J. A. Sloane

From Johannes W. Meijer, Jun 12 2010: (Start)
The terms of this sequence can be constructed with the terms of sequence A087130.
For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484.  (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Cf. A041047, A041018, A041046, A041090, A041150, A041226, A041318, A041426, A041550, A010484.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Oct 28 2013 *)
LinearRecurrence[ {0,0,0,0,140,0,0,0,0,1},{5,11,16,27,70,727,1524,2251,3775,9801},30] (* Harvey P. Dale, Jun 10 2021 *)

a(5*n) = A087130(3*n+1), a(5*n+1) = (A087130(3*n+2) - A087130(3*n+1))/2, a(5*n+2) = ( A087130(3*n+2) + A087130(3*n+1))/2, a(5*n+3) = A087130(3*n+2) and a(5*n+4) = A087130(3*n+3)/2. - Johannes W. Meijer, Jun 12 2010

G.f.: (5 + 11*x + 16*x^2 + 27*x^3 + 70*x^4 + 27*x^5 - 16*x^6 + 11*x^7 - 5*x^8 + x^9)/(1 - 140*x^5 - x^10) - Peter J. C. Moses, Jul 29 2013

A041047 Denominators of continued fraction convergents to sqrt(29).

Original entry on oeis.org

1, 2, 3, 5, 13, 135, 283, 418, 701, 1820, 18901, 39622, 58523, 98145, 254813, 2646275, 5547363, 8193638, 13741001, 35675640, 370497401, 776670442, 1147167843, 1923838285, 4994844413, 51872282415

Offset: 0

N. J. A. Sloane

The terms of this sequence can be constructed with the terms of sequence A052918.

For the terms of the periodical sequence of the continued fraction for sqrt(29) see A010128. We observe that its period is five. The decimal expansion of sqrt(29) is A010484. - Johannes W. Meijer, Jun 12 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,140,0,0,0,0,1).

Cf. A041046.

Cf. A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

I:=[1, 2, 3, 5, 13, 135, 283, 418, 701, 1820]; [n le 10 select I[n] else 140*Self(n-5)+Self(n-10): n in [1..50]]; // Vincenzo Librandi, Dec 10 2013

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[29],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[29], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

a(5*n) = A052918(3*n), a(5*n+1) = (A052918(3*n+1) - A052918(3*n))/2, a(5*n+2) = (A052918(3*n+1) + A052918(3*n))/2, a(5*n+3) = A052918(3*n+1) and a(5*n+4) = A052918(3*n+2)/2. - Johannes W. Meijer, Jun 12 2010
G.f.: (1 + 2*x + 3*x^2 + 5*x^3 + 13*x^4 - 5*x^5 + 3*x^6 - 2*x^7 + x^8)/(1 - 140*x^5 - x^10). - Peter J. C. Moses, Jul 29 2013
a(n) = 140*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 10 2013

A010538 Decimal expansion of square root of 87.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {3, 18} repeated. - Harry J. Smith, Jun 10 2009

			9.3273790530888150455544755423205569832762406...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2.

Cf. A040077 (continued fraction).

Cf. A002194, A010484.

RealDigits[N[87^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)

from math import isqrt
def aupton(nn): return list(map(int, str(isqrt(87 * 10**(2*nn)))))[:nn]
print(aupton(100)) # Michael S. Branicky, Sep 03 2021

Equals A002194 * A010484. - R. J. Mathar, Jun 08 2020

A176910 Decimal expansion of sqrt(145).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Apr 28 2010

Continued fraction expansion of sqrt(145) is 12 followed by A010863.

			sqrt(145) = 12.04159457879229548012...

Index entries for algebraic numbers, degree 2.

Cf. A002163 (decimal expansion of sqrt(5)), Cf. A010484 (decimal expansion of sqrt(29)), A176907 (decimal expansion of (9+sqrt(145))/16), A176908 (decimal expansion of (7+sqrt(145))/16), A010863 (all 24's sequence).

RealDigits[Sqrt[145],10,120][[1]] (* Harvey P. Dale, Oct 15 2023 *)

A374359 a(1) = 2, a(n) = 5 for n > 1.

Original entry on oeis.org

2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5

Offset: 1

Stefano Spezia, Jul 06 2024

Decimal expansion of 23/9, which is an approximation of the 5th root of 109 (A374357).

Simple continued fraction expansion of (1 + sqrt(29))/14 = (1 + A010484)/14.

			2.555555555555555555555555555555555555555...

Greg Martin and Winnie Miao, abc triples, arXiv:1409.2974 [math.NT], 2014. See p. 5.
Index entries for linear recurrences with constant coefficients, signature (1).

Cf. A374357 (decimal expansion of the 5th root of 109), A374358 (continued fraction of the 5th root of 109).
Cf. A010484.
Essentially the same as A021022 and A010716.

Mathematica
```
LinearRecurrence[{1},{2,5},100]
```

G.f.: x*(2 + 3*x)/(1 - x).
a(n) = a(n-1) for n > 2.
E.g.f.: 5*exp(x) - 3*x - 5.

A010128 Continued fraction for sqrt(29).

Original entry on oeis.org

5, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10, 2, 1, 1, 2, 10

Offset: 0

N. J. A. Sloane

			5.385164807134504031250710491... = 5 + 1/(2 + 1/(1 + 1/(1 + 1/(2 + ...)))). - _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 (0,0,0,0,1).

Cf. A010484 (Decimal expansion). - Harry J. Smith, Jun 04 2009

ContinuedFraction[Sqrt[29], 300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{5},120,{10,2,1,1,2}] (* Harvey P. Dale, Apr 18 2017 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(29)); for (n=0, 20000, write("b010128.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

G.f.: (5 + 2*x + x^2 + x^3 + 2*x^4 + 5*x^5)/(1 - x^5). - Stefano Spezia, Jul 26 2025

A085551 Decimal expansion of (sqrt(29)-5)/2.

Original entry on oeis.org

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

Offset: 0

Cino Hilliard, Jul 04 2003

			0.19258240356725201562535524577...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000

Cf. A010484 (sqrt(29)), A052918.

(sqrt(29)-5)/2 \\ Michel Marcus, Jan 16 2020

Equals Sum_{n>=1}(-1)^(n-1)/(A052918(n)*A052918(n+1)). - Vladimir Shevelev, Feb 23 2013

A178593 Decimal expansion of (7 + 5*sqrt(29))/26.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, May 31 2010

Continued fraction expansion of (7+5*sqrt(29))/26 is A084101.

			(7+5*sqrt(29))/26 = 1.30483938598740462139...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A010484 (decimal expansion of sqrt(29)), A084101 (repeat 1, 3, 3, 1).

SetDefaultRealField(RealField(100)); (7+5*Sqrt(29))/26; // G. C. Greubel, Jan 29 2019

RealDigits[(7+5*Sqrt[29])/26,10,120][[1]] (* Harvey P. Dale, Apr 09 2015 *)

default(realprecision, 100); (7+5*sqrt(29))/26 \\ G. C. Greubel, Jan 29 2019

numerical_approx((7+5*sqrt(29))/26, digits=100) # G. C. Greubel, Jan 29 2019

A367454 Decimal expansion of (-1 + sqrt(29))/14 = 1/A223140.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Jan 05 2024

c^n = 7*A(-(n+1)) + A(-n)*phi29, for n >= 0, where phi29 = A223140, and A(-n) = A015442(-n) = sqrt(-7)^(-(n+1))*S(-(n+1), 1/sqrt(-7)) = -(i/sqrt(7))^(n+1)*S(n-1, i/sqrt(7)), with i = sqrt(-1) and the S-Chebyshev polynomials (see A049310), where S(-n, x) = -S(n-2, x), for n >= 1, and S(n, -x) = (-1)^n*S(n, x).

			c = 0.3132260576524645736607650351100235397353657258317706312628...

Index entries for sequences related to Chebyshev polynomials.

Cf. A010484, A015442, A049310, A223140.

Flatten[First[RealDigits[(-1 + Sqrt[29])/14,10,96]]] (* Stefano Spezia, Jan 05 2024 *)

c = 1/phi29 = (-1 + phi(29))/7, with phi29 = A223140.

cp's OEIS Frontend

A248256 Egyptian fraction representation of sqrt(29) (A010484) using a greedy function.

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A041046 Numerators of continued fraction convergents to sqrt(29).

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A041047 Denominators of continued fraction convergents to sqrt(29).

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A010538 Decimal expansion of square root of 87.

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A176910 Decimal expansion of sqrt(145).

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A374359 a(1) = 2, a(n) = 5 for n > 1.

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A010128 Continued fraction for sqrt(29).

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A085551 Decimal expansion of (sqrt(29)-5)/2.

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