A010464 -id:A010464 - cp's OEIS frontend

A248236 Egyptian fraction representation of sqrt(6) (A010464) using a greedy function.

2, 3, 9, 199, 49572, 30799364495, 1408429952507887000310, 3677260735023142918878205127156519291320765, 102293202370266874495262346614859561910266026424997387777849999466054887759064682698213

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[ 6]]

A022840 Beatty sequence for sqrt(6).

Original entry on oeis.org

2, 4, 7, 9, 12, 14, 17, 19, 22, 24, 26, 29, 31, 34, 36, 39, 41, 44, 46, 48, 51, 53, 56, 58, 61, 63, 66, 68, 71, 73, 75, 78, 80, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 110, 112, 115, 117, 120, 122, 124, 127, 129, 132, 134, 137, 139, 142, 144, 146

Offset: 1

Clark Kimberling

nonn

Complement of A138235; a(n) = A138236(A138235(n)) and A138236(a(n)) = A138235(n). - Reinhard Zumkeller, Mar 07 2008

Numbers k such that A248515(k+1) = A248515(k) + 1 = 1 + least number h such that 1 - h*sin(1/h) < 1/n^2. The difference sequence of A248515 is (0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, ...), so that A138235 = (1, 3, 5, 6, 8, ...) and A022840 = (2, 4, 7, 9, 12, 14, ...). - Clark Kimberling, Jun 16 2015

Iain Fox, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Cf. A010464 (sqrt(6)), A138235 (complement), A248515.

a022840 = floor . (* sqrt 6) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

[Floor(n*Sqrt(6)): n in [1..60]]; // Vincenzo Librandi, Jun 17 2015

Table[Floor[n Sqrt[6]], {n, 70}] (* Vincenzo Librandi, Jun 17 2015 *)

a(n) = floor(n*sqrt(6)) \\ Iain Fox, Nov 20 2017

A086180 Decimal expansion of 1 + sqrt(6).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 11 2003

Onset of 4-cycle in the logistic equation.

			3.4494897427831780981972840747058913919659474806567...

Gian Italo Bischi, Rosa Carini, Laura Gardini, and Paolo Tenti, Sulle orme del caos: comportamenti complessi in modelli matematici semplici. Bruno Mondadori (Milano), 2004. See pp. 83, 92, 95. (In Italian)
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.9, p. 66.

Andy Burbanks, Period Doubling: Iteration of the Logistic Map.
Kurt Cogswell, WebDiagram Applet.
Michael Cross, The Period Doubling Route To Chaos.
Blair Fraser, Iterated Maps.
Alan Hood, Logistic Equation.
Pete Kernan, Logistic Map(Cobweb Construction).
Pete Kernan, Logistic Map(Time Series).
Oliver Knill, Attractor of quadratic map.
Daniel Marthaler, Logistic Map.
John McDonald, Iteration versus time for the Logistic map.
Frank Wattenberg, An Applet in a Window--Cobweb Diagrams.
Eric Weisstein's World of Mathematics, Logistic Map.
Index entries for algebraic numbers, degree 2.

Cf. A010464, A086178.

RealDigits[1+Sqrt[6],10,120][[1]] (* Harvey P. Dale, Jul 28 2012 *)

1+sqrt(6) \\ Stefano Spezia, Nov 17 2024

Equals A010464 plus 1. - R. J. Mathar, Sep 12 2008

A115754 Decimal expansion of sqrt(3/2).

Original entry on oeis.org

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

Offset: 1

Eric Desbiaux, Jul 30 2008

Coordinate of a control point for a degree-5 integration formula for 7 points over the unit circle. [Stroud & Secrest]
Also real and imaginary part of sqrt(-3i). - Alonso del Arte, Dec 11 2012
Area of the quadrilateral obtained when slicing a unit cube with a plane passing through opposite vertices and the middle of opposite edges. See CNRS link. - Michel Marcus, Mar 26 2016
Positive zero of the Hermite polynomial of degree 3. - A.H.M. Smeets, Jun 02 2025

			1.2247448713915890490986420373529456959829737403283350642163...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000
Ana Rechtman, Mars 2016, 3e défi, Images des Mathématiques, CNRS, 2016.
A. H. Stroud and D. Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Index entries for algebraic numbers, degree 2

Cf. A382713 (continued fraction), A068388 (Engel expansion).

Cf. A010464 (double), A187110 (half), A157697 (reciprocal).

RealDigits[Sqrt[3/2], 10, 105][[1]] (* Alonso del Arte, Dec 11 2012 *)

Equals 2*A187110.
Equals Sum_{k>=0} binomial(1/2, k)/2^k. - Bruno Berselli, Sep 11 2015
From Amiram Eldar, Aug 02 2020: (Start)
Equals Product_{k>=0} (1 + (-1)^k/(6*k + 3)).
Equals Sum_{k>=0} binomial(2*k,k)/12^k.
Equals 1 + Sum_{k>=1} (2*k - 1)!!/((2*k)!! * 3^k). (End)
Equals A010464/2. - R. J. Mathar, Feb 23 2021

A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron with edge length 1.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Aug 30 2007

Volume of a regular octahedron: V = ((sqrt(2))/3)* a^3, where 'a' is the edge.

			0.471404520791031682933896...

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, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A020829 (regular tetrahedron volume), A102208 (regular icosahedron volume), A102769 (regular dodecahedron volume).

Cf. A179587.

RealDigits[Sqrt[2]/3,10,120][[1]] (* Harvey P. Dale, May 27 2012 *)

PARI
```
sqrt(2)/3 \\ G. C. Greubel, Jul 06 2017
```

Equals A002193/3 = A010464/A010482. - R. J. Mathar, Dec 11 2009

More digits from R. J. Mathar, Dec 11 2009

A010480 Decimal expansion of square root of 24.

Original entry on oeis.org

4, 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, 3, 7, 1, 8

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 4 followed by {1, 8} repeated. - Harry J. Smith, Jun 03 2009
The solution to the Lane-Emden equation for a sphere of polytropic index n = 0. - Arkadiusz Wesolowski, Nov 11 2014
Using Bretschneider's formula this is the maximum area of a quadrilateral with side lengths of 1,2,3 and 4. - Scott R. Shannon, Jan 06 2020

			4.898979485566356196394568149411782783931894961313340256865385134501920....

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

Cf. A040019 (continued fraction). - Harry J. Smith, Jun 03 2009

Cf. A010464, A090654.

RealDigits[N[Sqrt[24],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(24); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010480.txt", n, " ", d));  \\ Harry J. Smith, Jun 03 2009

Equals 2*A010464. - R. J. Mathar, Jan 14 2021

Equals 4 + Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 8^k). - Amiram Eldar, May 06 2022

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A010485 Decimal expansion of square root of 30.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {2, 10} repeated. - Harry J. Smith, Jun 04 2009

			5.477225575051661134569697828008021339527446949979832542268944....

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

Cf. A040024, continued fraction. - Harry J. Smith, Jun 04 2009

RealDigits[N[Sqrt[30], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(30); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010485.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Equals A010464*A002163 = 1/A020787. - R. J. Mathar, Dec 17 2024

A020763 Decimal expansion of 1/sqrt(6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

Radius of the inscribed sphere (tangent to all faces) in a regular octahedron with unit edge. - Stanislav Sykora, Nov 21 2013

			0.408248290463863016366214012450981898660991246776111688072115427875...

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, Platonic solid.
Index entries for algebraic numbers, degree 2.

Cf. A010464, A157697.

Cf. Platonic solids in radii: A020781 (tetrahedron), A179294 (icosahedron), A237603 (dodecahedron). - Stanislav Sykora, Feb 25 2014

evalf(1/sqrt(6)); # Michal Paulovic, Dec 09 2022

RealDigits[1/Sqrt[6], 10, 100][[1]] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)
realDigitsRecip[Sqrt[6]] (* The realDigitsRecip program is at A021200 *) (* Harvey P. Dale, Jan 10 2025 *)

From Michal Paulovic, Dec 09 2022: (Start)
Equals A157697/2 = A010503 * A020760 = 1/A010464.
Equals [0, 2; 2, 4] (periodic continued fraction expansion). (End)

A041007 Denominators of continued fraction convergents to sqrt(6).

Original entry on oeis.org

1, 2, 9, 20, 89, 198, 881, 1960, 8721, 19402, 86329, 192060, 854569, 1901198, 8459361, 18819920, 83739041, 186298002, 828931049, 1844160100, 8205571449, 18255302998, 81226783441, 180708869880

Offset: 0

N. J. A. Sloane

sqrt(6) = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721), ...; where sqrt(6) = 2.4494897427... and the sum of the first 5 terms of this series = 2.449489737... - Gary W. Adamson, Dec 21 2007
sqrt(6) = 2 + continued fraction [2, 4, 2, 4, 2, 4, ...] = 4/2 + 4/9 + 4/(9*89) + 4/(89*881) + 4/(881*8721) + ... - Gary W. Adamson, Dec 21 2007
Interspersion of 2 sequences, A072256 and 2*A004189. - Gerry Martens, Jun 10 2015
For n > 0, a(n) equals the permanent of the n X n tridiagonal matrix with the main diagonal alternating sequence [2, 4, 2, 4, ...] and 1's along the superdiagonal and the subdiagonal. - Rogério Serôdio, Apr 01 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A010464, A041006.

Cf. A072256, A004189.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)

G.f.: (1+2*x-x^2)/(1-10*x^2+x^4). - Colin Barker, Dec 31 2011
From Rogério Serôdio, Apr 01 2018: (Start)
Recurrence formula: a(n) = (3 + (-1)^n)*a(n-1) + a(n-2), a(0) = 1, a(1) = 2.
Some properties:
(1) a(n)^2 - a(n-2)^2 = (3+(-1)^n)*a(2*n-1), for n > 1;
(2) a(2*n+1) = a(n)*(a(n+1) + a(n-1)), for n > 0;
(3) a(2*n) = A142239(2*n), for n >= 0;
(4) a(2*n+1) = A041007(2*n+1)/2, for n >= 0;
(5) a(2*n-1)*A142239(2*n+1) = a(n)^2 - 1, for n > 0;
(6) a(2*n) = a(n)*A142239(n) + a(n-1)*A142239(n-1), for n > 0;
(7) Sum_{k=0..n} a(2*k+1)*(A142239(2*k) + A142239(2*(k+1))) = Sum_{k=0..n} a(3+4*k);
(8) Sum_{k=0..n} (a(2*k-1) + a(2*k+1))*A142239(2*k) = Sum_{k=0..n} A142239(3+4*k). (End)
a(n) = ((2 + sqrt(6))^(n+1) - (2 - sqrt(6))^(n+1))/(sqrt(6) * 2^(ceiling(n/2) + 1)). - Robert FERREOL, Oct 14 2024

A135611 Decimal expansion of sqrt(2) + sqrt(3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 03 2008

From Alexander R. Povolotsky, Mar 04 2008: (Start)
The value of sqrt(2) + sqrt(3) ~= 3.146264369941972342329135... is "close" to Pi. [See Borel 1926. - Charles R Greathouse IV, Apr 26 2014] We can get a better approximation by solving the equation: (2-x)^(1/(2+x)) + (3-x)^(1/(2+x)) = Pi.
Olivier Gérard finds that x is 0.00343476569746030039595770020414255107204742044644777... (End)
Another approximation to Pi is (203*sqrt(2)+ 197*sqrt(3))/200 = 3.1414968... - Alexander R. Povolotsky, Mar 22 2008
Shape of a sqrt(8)-extension rectangle; see A188640.  - Clark Kimberling, Apr 13 2011
This number is irrational, as instinct would indicate. Niven (1961) gives a proof of irrationality that requires first proving that sqrt(6) is irrational. - Alonso del Arte, Dec 07 2012
An algebraic integer of degree 4: largest root of x^4 - 10*x^2 + 1. - Charles R Greathouse IV, Sep 13 2013
Karl Popper considers whether this approximation to Pi might have been known to Plato, or even conjectured to be exact. - Charles R Greathouse IV, Apr 26 2014

			3.14626436994197234232913506571557044551247712918732870...

Emile Borel, Space and Time (1926).
Ivan Niven, Numbers: Rational and Irrational. New York: Random House for Yale University (1961): 44.
Ian Stewart & David Tall, Algebraic Number Theory and Fermat's Last Theorem, 3rd Ed. Natick, Massachusetts: A. K. Peters (2002): 44.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Susan Landau, Simplification of nested radicals, SIAM Journal on Computing 21.1 (1992): 85-110. See page 85. [Do not confuse this paper with the short FOCS conference paper with the same title, which is only a few pages long.]
Burkard Polster, Irrational roots, Mathologer video (2018)
Karl Popper, The Open Society and Its Enemies, 1962.
Index entries for algebraic numbers, degree 4

Cf. A188640, A089078, A002193, A002194, A010464, A340616.

SetDefaultRealField(RealField(100)); Sqrt(2) + Sqrt(3); // G. C. Greubel, Nov 20 2018

evalf(add(sqrt(ithprime(i)), i=1..2), 118);  # Alois P. Heinz, Jun 13 2022

r = 8^(1/2); t = (r + (4 + r^2)^(1/2))/2; FullSimplify[t]
N[t, 130]
RealDigits[N[t, 130]][[1]] (* A135611 *)
ContinuedFraction[t, 120]  (* A089078 *)
RealDigits[Sqrt[2] + Sqrt[3], 10, 100][[1]] (* G. C. Greubel, Oct 22 2016 *)

sqrt(2)+sqrt(3) \\ Charles R Greathouse IV, Sep 13 2013

numerical_approx(sqrt(2)+sqrt(3), digits=100) # G. C. Greubel, Nov 20 2018

Sqrt(2)+sqrt(3) = sqrt(5+2*sqrt(6)). [Landau, p. 85] - N. J. A. Sloane, Aug 27 2018
Equals 1/A340616. - Hugo Pfoertner, May 08 2024
Equals Product_{k>=0} (((4*k + 1)*(12*k + 11))/((4*k + 3)*(12*k + 1)))^(-1)^k. - Antonio Graciá Llorente, May 22 2024

cp's OEIS Frontend

A248236 Egyptian fraction representation of sqrt(6) (A010464) using a greedy function.

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A022840 Beatty sequence for sqrt(6).

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A086180 Decimal expansion of 1 + sqrt(6).

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A115754 Decimal expansion of sqrt(3/2).

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A131594 Decimal expansion of sqrt(2)/3, the volume of a regular octahedron with edge length 1.

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A010480 Decimal expansion of square root of 24.

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A010485 Decimal expansion of square root of 30.

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