A010478 -id:A010478 - cp's OEIS frontend

A248250 Egyptian fraction representation of sqrt(22) (A010478) using a greedy function.

4, 2, 6, 43, 2028, 5477762, 40063230724280, 10039617492048087897098971783, 598943577818423089223821862011302605314284839297545338532, 451273778419286656581820003198742640276389207705020449590295850757882195737121214614786626350432663721793231915121

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[ 22]]

A010126 Continued fraction for sqrt(22).

Original entry on oeis.org

4, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2, 4, 2, 1, 8, 1, 2

Offset: 0

N. J. A. Sloane

			4.690415759823429554565630113... = 4 + 1/(1 + 1/(2 + 1/(4 + 1/(2 + ...)))). - _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 (0,0,0,0,0,1).

Cf. A041034/A041035 (convergents), A248250 (Egyptian fraction), A010478 (decimal expansion).

ContinuedFraction[Sqrt[22],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1,2,4,2,1}] (* Harvey P. Dale, Jul 02 2019 *)

{ allocatemem(932245000); default(realprecision, 18000); x=contfrac(sqrt(22)); for (n=0, 20000, write("b010126.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2^e) = 2, a(3^e) = 4, and a(p^e) = 1 for p >= 5.
Dirichlet g.f.: zeta(s) * (1 + 1/2^s) * (1 + 1/3^(s-1)). (End)
G.f.: (4 + x + 2*x^2 + 4*x^3 + 2*x^4 + x^5 + 4*x^6)/(1 - x^6). - Stefano Spezia, Jul 26 2025

A041034 Numerators of continued fraction convergents to sqrt(22).

Original entry on oeis.org

4, 5, 14, 61, 136, 197, 1712, 1909, 5530, 24029, 53588, 77617, 674524, 752141, 2178806, 9467365, 21113536, 30580901, 265760744, 296341645, 858444034, 3730117781, 8318679596, 12048797377, 104709058612

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 (0,0,0,0,0,394,0,0,0,0,0,-1).

Cf. A010478, A041035.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[22],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
CoefficientList[Series[- (x^11 - 4 x^10 + 5 x^9 - 14 x^8 + 61 x^7 - 136 x^6 - 197 x^5 - 136 x^4 - 61 x^3 - 14 x^2 - 5 x - 4)/(x^12 - 394 x^6 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)
LinearRecurrence[{0,0,0,0,0,394,0,0,0,0,0,-1},{4,5,14,61,136,197,1712,1909,5530,24029,53588,77617},30] (* Harvey P. Dale, Mar 14 2017 *)

a(n) = 394*a(n-6)-a(n-12). G.f.: -(x^11 -4*x^10 +5*x^9 -14*x^8 +61*x^7 -136*x^6 -197*x^5 -136*x^4 -61*x^3 -14*x^2 -5*x -4)/(x^12-394*x^6+1). [Colin Barker, Jul 16 2012]

A041035 Denominators of continued fraction convergents to sqrt(22).

Original entry on oeis.org

1, 1, 3, 13, 29, 42, 365, 407, 1179, 5123, 11425, 16548, 143809, 160357, 464523, 2018449, 4501421, 6519870, 56660381, 63180251, 183020883, 795263783, 1773548449, 2568812232, 22324046305, 24892858537

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 (0,0,0,0,0,394,0,0,0,0,0,-1).

Cf. A010478, A041034.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[22],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 17 2011 *)
LinearRecurrence[{0,0,0,0,0,394,0,0,0,0,0,-1 }, {1, 1, 3, 13, 29, 42, 365, 407, 1179, 5123, 11425, 16548}, 50] (* Stefano Spezia, Sep 30 2018 *)
CoefficientList[Series[-(x^10 - x^9 + 3*x^8 - 13*x^7 + 29*x^6 - 42*x^5 - 29*x^4 - 13*x^3 - 3*x^2 - x- 1)/(x^12 - 394*x^6 + 1), {x, 0, 50}], x] (* Stefano Spezia, Sep 30 2018 *)

vector(26, i, contfracpnqn(contfrac(sqrt(22), i))[2,1]) \\ Arkadiusz Wesolowski, Sep 29 2018

Vec(-(x^10 - x^9 + 3*x^8 - 13*x^7 + 29*x^6 - 42*x^5 - 29*x^4 - 13*x^3 - 3*x^2 - x - 1)/(x^12 - 394*x^6 + 1) + O(x^50)) \\ Stefano Spezia, Sep 30 2018

From Colin Barker, Jul 16 2012: (Start)
a(n) = 394*a(n-6) - a(n-12).
G.f.: -(x^10 - x^9 + 3*x^8 - 13*x^7 + 29*x^6 - 42*x^5 - 29*x^4 - 13*x^3 - 3*x^2 - x - 1)/(x^12 - 394*x^6 + 1). (End)

A020779 Decimal expansion of 1/sqrt(22).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.21320071635561043429843773243383937...

Ivan Panchenko, Table of n, a(n) for n = 0..1000

Cf. A010478 (reciprocal).

RealDigits[N[1/Sqrt[22],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

A377276 Decimal expansion of the circumradius of a truncated tetrahedron with unit edge length.

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Oct 23 2024

			1.17260393995585738864140752838611657014705708835...

Eric Weisstein's World of Mathematics, Truncated Tetrahedron.
Wikipedia, Truncated tetrahedron.

Cf. A377274 (surface area), A377275 (volume), A093577 (midradius), A377277 (Dehn invariant).
Cf. A187110 (analogous for a regular tetrahedron).
Cf. A010478.

First[RealDigits[Sqrt[22]/4, 10, 100]] (* or *)
First[RealDigits[PolyhedronData["TruncatedTetrahedron", "Circumradius"], 10, 100]]

Equals sqrt(22)/4 = A010478/4.

A017970 Powers of sqrt(22) rounded down.

Original entry on oeis.org

1, 4, 22, 103, 484, 2270, 10648, 49943, 234256, 1098758, 5153632, 24172676, 113379904, 531798888, 2494357888, 11699575548, 54875873536, 257390662067, 1207269217792, 5662594565481, 26559922791424

Offset: 0

N. J. A. Sloane

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..700

Cf. A010478 (sqrt(22)).

[Floor(Sqrt(22^n)): n in [0..30]]; // Vincenzo Librandi, Jun 24 2011

Floor[(Sqrt[22])^Range[0,30]] (* Harvey P. Dale, Jul 16 2020 *)

a(n)=sqrtint(22^n) \\ Charles R Greathouse IV, Nov 18 2011

a(n) = floor(sqrt(22^n)). - Vincenzo Librandi, Jun 24 2011

cp's OEIS Frontend

A248250 Egyptian fraction representation of sqrt(22) (A010478) using a greedy function.

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A010126 Continued fraction for sqrt(22).

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A041034 Numerators of continued fraction convergents to sqrt(22).

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A041035 Denominators of continued fraction convergents to sqrt(22).

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A020779 Decimal expansion of 1/sqrt(22).

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A377276 Decimal expansion of the circumradius of a truncated tetrahedron with unit edge length.

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A017970 Powers of sqrt(22) rounded down.

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