A010480 -id:A010480 - cp's OEIS frontend

A248252 Egyptian fraction representation of sqrt(24) (A010480) using a greedy function.

4, 2, 3, 16, 318, 667493, 520599832812, 1406502882894868771562029, 5482100301108869539661068478608291549480253128390, 195012261486920753888173091467257385308263858947121366558714224718185929485569758493733677353323155

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[ 24]]

A090654 Decimal expansion of 4 + 2*sqrt(6).

Original entry on oeis.org

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

Offset: 1

Felix Tubiana, Feb 05 2004

Equals n +n/(n +n/(n +n/(n +....))) for n = 8. See also A090388. - Stanislav Sykora, Jan 23 2014

			8.898979485566356196394568149...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2

Cf. A010480, A176213.
Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090488 (n=4), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Essentially the same as A010480.

RealDigits[4 + 2*Sqrt[6], 10, 50][[1]] (* G. C. Greubel, Jul 03 2017 *)

4 + 2*sqrt(6) \\ G. C. Greubel, Jul 03 2017

A020781 Decimal expansion of 1/sqrt(24).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to the faces) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			1/sqrt(24) = 0.20412414523193150818310700622549094933... . - _Vladimir Joseph Stephan Orlovsky_, May 30 2010

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 450.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. Platonic solids inradii: A020763 (octahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

Cf. A010464, A010480, A020763, A020853, A021028, A187110, A244980.

RealDigits[N[1/Sqrt[24],200]] (* Vladimir Joseph Stephan Orlovsky, May 30 2010 *)

Equals A010464/12. - Stefano Spezia, Jan 26 2025

Equals 1/A010480 = A020763/2 = 2*A020853 = A187110/3 = A244980/Pi. - Hugo Pfoertner, Jan 26 2025

A041038 Numerators of continued fraction convergents to sqrt(24).

Original entry on oeis.org

4, 5, 44, 49, 436, 485, 4316, 4801, 42724, 47525, 422924, 470449, 4186516, 4656965, 41442236, 46099201, 410235844, 456335045, 4060916204, 4517251249, 40198926196, 44716177445, 397928345756

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 (0,10,0,-1).

Cf. A010480, A041039.

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[24],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011*)
Numerator[Convergents[Sqrt[24], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

a(2n) = 2*A041006(2n) ; a(2n-1) = A041006(2n-1) = A001079(n). [From M. F. Hasler, Feb 13 2009]

G.f.: (4+5*x+4*x^2-x^3)/(1-10*x^2+x^4)

A040019 Continued fraction for sqrt(24).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Decimal expansion of 23/55. - R. J. Mathar, Aug 25 2025

			4.898979485566356196394568149... = 4 + 1/(1 + 1/(8 + 1/(1 + 1/(8 + ...)))). - _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 (0,1).

Cf. A010480 (decimal expansion), A010689.

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

ContinuedFraction[Sqrt[24],300] (* Vladimir Joseph Stephan Orlovsky, Mar 05 2011 *)
PadRight[{4},120,{8,1}] (* Harvey P. Dale, Oct 24 2022 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(24)); for (n=0, 20000, write("b040019.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 03 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2^e) = 8, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 7/2^s). (End)
G.f.: (4 + x + 4*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A041039 Denominators of continued fraction convergents to sqrt(24).

Original entry on oeis.org

1, 1, 9, 10, 89, 99, 881, 980, 8721, 9701, 86329, 96030, 854569, 950599, 8459361, 9409960, 83739041, 93149001, 828931049, 922080050, 8205571449, 9127651499, 81226783441, 90354434940, 804062262961

Offset: 0

N. J. A. Sloane

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 8 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Eric W. Weisstein, MathWorld: Lehmer Number
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A010480, A041038, A002530.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[24],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Denominator[Convergents[Sqrt[24],30]] (* or *) LinearRecurrence[{0,10,0,-1},{1,1,9,10},30] (* Harvey P. Dale, Apr 12 2022 *)

G.f.: (1+x-x^2)/(1-10*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(2) + sqrt(3) and beta = sqrt(2) - sqrt(3) be the roots of the equation x^2 - sqrt(8)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 8 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 8*a(2*n) + a(2*n - 1). (End)

A248672 Decimal expansion of the solution to the Lane-Emden equation for a sphere of polytropic index n = 2.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Oct 11 2014

			2.4110460120968937836484427446714...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 2.14, pp. 338-339.

Vassilis S. Geroyannis and Vasileios G. Karageorgopoulos, A parallel code for multiprecision computations of the Lane-Emden differential equation, arXiv:1604.08019 [astro-ph.SR], 2016.
Wikipedia, Lane-Emden equation.

Cf. A010480 (n=0), A000796 (n=1), A248673 (n=3), A248674 (n=4), A002194 (n=5).

More terms from Geroyannis and Karageorgopoulos (2016) added by Amiram Eldar, May 15 2021

A248673 Decimal expansion of the solution to the Lane-Emden equation for a sphere of polytropic index n = 3.

Original entry on oeis.org

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

Offset: 1

Arkadiusz Wesolowski, Oct 11 2014

			2.0182359509662284028128131700579...

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 2.14, pp. 338-339.

Vassilis S. Geroyannis and Vasileios G. Karageorgopoulos, A parallel code for multiprecision computations of the Lane-Emden differential equation, arXiv:1604.08019 [astro-ph.SR], 2016.
Wikipedia, Lane-Emden equation.

Cf. A010480 (n=0), A000796 (n=1), A248672 (n=2), A248674 (n=4), A002194 (n=5).

More terms from Geroyannis and Karageorgopoulos (2016) added by Amiram Eldar, May 15 2021

A017976 Powers of sqrt(24) rounded down.

Original entry on oeis.org

1, 4, 24, 117, 576, 2821, 13824, 67723, 331776, 1625363, 7962624, 39008731, 191102976, 936209559, 4586471424, 22469029417, 110075314176, 539256706015, 2641807540224, 12942160944371, 63403380965376

Offset: 0

N. J. A. Sloane

nonn

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

Cf. A010480 (sqrt(24)).

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

Floor[(Sqrt[24])^Range[0,20]] (* Harvey P. Dale, Mar 01 2015 *)

a(n)=sqrtint(24^n) \\ Charles R Greathouse IV, Nov 18 2011

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

A200728 Decimal expansion of the circumradius of cyclic quadrilateral with sides 1, 2, 3, 4.

Original entry on oeis.org

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

Offset: 1

Zak Seidov, Nov 21 2011

Area is K = sqrt(24) = 4.89897948556635619639... (decimal expansion in A010480).

			2.0026024734496526299517...

Zak Seidov, Figure showing polygon.
Index entries for algebraic numbers, degree 2

Cf. A090654, A010480, A200113, A200257.

RealDigits[Sqrt[385/96], 10, 120][[1]] (* Amiram Eldar, Jun 27 2023 *)

sqrt(385/96) \\ Charles R Greathouse IV, Dec 28 2011

R = (385/96)^(1/2).

cp's OEIS Frontend

A248252 Egyptian fraction representation of sqrt(24) (A010480) using a greedy function.

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A090654 Decimal expansion of 4 + 2*sqrt(6).

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A020781 Decimal expansion of 1/sqrt(24).

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A041038 Numerators of continued fraction convergents to sqrt(24).

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A040019 Continued fraction for sqrt(24).

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A041039 Denominators of continued fraction convergents to sqrt(24).

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A248672 Decimal expansion of the solution to the Lane-Emden equation for a sphere of polytropic index n = 2.

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A248673 Decimal expansion of the solution to the Lane-Emden equation for a sphere of polytropic index n = 3.

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