A010488 -id:A010488 - cp's OEIS frontend

A248260 Egyptian fraction representation of sqrt(33) (A010488) using a greedy function.

5, 2, 5, 23, 923, 1039448, 1349594009502, 1841990944227649463764190, 5531888379621714420992617902281239594988386275117, 172423874327527416450254906621893256497583527925050132860644029730203113536215473159687066655835408

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[ 33]]

A235162 Decimal expansion of (sqrt(33) + 1) / 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 06 2014

Solution of y^2 - y - 8 = 0.
Decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
The sequence with a(1) = 2 is decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
A basis for the integers of the real quadratic number field K(sqrt(33)) is
  <1, omega(33)>, where omega(33) = (1 + sqrt(33))/2. - Wolfdieter Lang, Feb 11 2020

			3.37228132326901432992530573410946465911013222899139618384993873528...

Christopher M. Conrey, Table of n, a(n) for n = 1..10000

Cf. A001622, A010488, A209927, A222132, A222134, A223140.

val = vpa((sqrt(sym(33))+1)/2,10001); list = char(val)-'0'; list = list([1,3:end-1]); % Christopher M. Conrey, Jan 26 2022

Mathematica
```
RealDigits[(1 + Sqrt[33])/2, 10, 130]
```

A010131 Continued fraction for sqrt(33).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			5.74456264653802865985061146... = 5 + 1/(1 + 1/(2 + 1/(1 + 1/(10 + ...)))). - _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 (0,0,0,1).

Cf. A010488 (decimal expansion).

ContinuedFraction[Sqrt[33],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
LinearRecurrence[{0,0,0,1},{5,1,2,1,10},100] (* or *) PadRight[{5},100,{10,1,2,1}] (* Harvey P. Dale, Dec 31 2017 *)

{ allocatemem(932245000); default(realprecision, 17000); x=contfrac(sqrt(33)); for (n=0, 20000, write("b010131.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 2, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 1/2^(2*s-3) + 1/2^s). (End)
G.f.: (5 + x + 2*x^2 + x^3 + 5*x^4)/(1 - x^4). - Stefano Spezia, Jul 26 2025

A041054 Numerators of continued fraction convergents to sqrt(33).

Original entry on oeis.org

5, 6, 17, 23, 247, 270, 787, 1057, 11357, 12414, 36185, 48599, 522175, 570774, 1663723, 2234497, 24008693, 26243190, 76495073, 102738263, 1103877703, 1206615966, 3517109635, 4723725601, 50754365645

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,46,0,0,0,-1).

Cf. A010488, A041055.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[33], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[33],30]] (* Harvey P. Dale, Oct 25 2011 *)
CoefficientList[Series[- (x^7 - 5 x^6 + 6 x^5 - 17 x^4 - 23 x^3 - 17 x^2 - 6 x - 5)/(x^8 - 46 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 23 2013 *)

a(n) = 46*a(n-4)-a(n-8). G.f.: -(x^7-5*x^6+6*x^5-17*x^4-23*x^3-17*x^2-6*x-5)/(x^8-46*x^4+1). [Colin Barker, Jul 16 2012]

A041055 Denominators of continued fraction convergents to sqrt(33).

Original entry on oeis.org

1, 1, 3, 4, 43, 47, 137, 184, 1977, 2161, 6299, 8460, 90899, 99359, 289617, 388976, 4179377, 4568353, 13316083, 17884436, 192160443, 210044879, 612250201, 822295080, 8835201001, 9657496081, 28150193163

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 (0,0,0,46,0,0,0,-1).

Cf. A010488, A041054.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[33], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[33], 30]] (* Vincenzo Librandi, Oct 23 2013 *)
LinearRecurrence[{0,0,0,46,0,0,0,-1},{1,1,3,4,43,47,137,184},30] (* Harvey P. Dale, Apr 24 2022 *)

a(n) = 46*a(n-4)-a(n-8). G.f.: -(x^2-x-1)*(x^4+4*x^2+1)/(x^8-46*x^4+1). [Colin Barker, Jul 16 2012]

A236290 Decimal expansion of (sqrt(33) - 1) / 2.

Original entry on oeis.org

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

Offset: 1

Jaroslav Krizek, Feb 07 2014

Decimal expansion of sqrt(8 - sqrt(8 - sqrt(8 - sqrt(8 - ... )))).
The sequence with a(1) = 3 is decimal expansion of sqrt(8 + sqrt(8 + sqrt(8 + sqrt(8 + ... )))).
A quadratic integer with minimal polynomial x^2 + x - 8. - Charles R Greathouse IV, Apr 21 2016
Triangular root of 4. - Stefano Spezia, Sep 05 2025

			2.37228132326901432992530573410946465911013222899139618384993873528...

Colin Foster, Triangular roots, Applied Probability Trust, pp. 8-9, 2012.
Index entries for algebraic numbers, degree 2.

Cf. A010488, A235162.

evalf((sqrt(33)-1)/2, 140);  # Alois P. Heinz, Sep 05 2025

Mathematica
```
RealDigits[(Sqrt[33] - 1)/2, 10, 130]
```

(sqrt(33)-1)/2 \\ Charles R Greathouse IV, Apr 21 2016

Equals A235162 - 1.

A357418 Decimal expansion of (207 - 33*sqrt(33))/32.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Sep 27 2022

This constant expresses the Turán density of the extension of the 4-graph formed by 3 disjoint edges. For the details about the construction of the extension, see Wu's article.

			0.54466977075765794452905692339922914058535...

Biao Wu, An irrational Turán density via hypergraph Lagrangian densities, The Electronic Journal of Combinatorics 29(3) (2022), #P3.62.

Cf. A010488.

First[RealDigits[N[(207 - 33 Sqrt[33])/32, 95]]]

Minimal polynomial: 16*x^2 - 207*x + 108. - Stefano Spezia, Aug 03 2025

cp's OEIS Frontend

A248260 Egyptian fraction representation of sqrt(33) (A010488) using a greedy function.

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A235162 Decimal expansion of (sqrt(33) + 1) / 2.

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A010131 Continued fraction for sqrt(33).

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A041054 Numerators of continued fraction convergents to sqrt(33).

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A041055 Denominators of continued fraction convergents to sqrt(33).

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A236290 Decimal expansion of (sqrt(33) - 1) / 2.

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Maple

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A357418 Decimal expansion of (207 - 33*sqrt(33))/32.

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