A010503 -id:A010503 - cp's OEIS frontend

A099972 Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

1, 5, 13, 45, 173, 8365, 73901, 204973, 467117, 991405, 5185709, 13574317, 80683181, 214900909, 1288642733, 3436126381, 7731093677, 16321028269, 33500897453, 67860635821, 136580112557, 686335926445, 1785847554221

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			1/sqrt(2) = 0.7071067811865475244008443621048490392848359376885... = 0.10110101000001001111001100110011111110011101111001100100100001000101100101111101100010011011 in binary.

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A010503, A099969, A099970, A099971, A099973, A099974.

d = 100; l = First[RealDigits[N[1/Sqrt[2], d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 18 2005 *)
Module[{rd=RealDigits[1/Sqrt[2],2,50][[1]],pos},pos=Flatten[Position[rd,1]];Table[ FromDigits[ Reverse[Take[rd,n]],2],{n,pos}]] (* Harvey P. Dale, Jul 29 2013 *)

More terms from Ryan Propper, Aug 18 2005

A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 09 2010

Dodecahedron: A three-dimensional figure with 12 faces, 20 vertices, and 30 edges.

Appears as a coordinate in a degree-7 quadrature formula on 12 points over the unit circle [Stroud & Secrest]. - R. J. Mathar, Oct 12 2011

			1.40125853844407354467667793532206799444393173977549286366084518639135...

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §12.4 Theorems and Formulas (Solid Geometry), p. 451.

Chai Wah Wu, Table of n, a(n) for n = 1..10001
A. H. Stroud and Don Secrest, Approximate integration formulas for certain spherically symmetric regions, Math. Comp. 17 (82) (1963) 105.
Wikipedia, Dodecahedron.
Wolfram Alpha, Dodecahedron.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A102208, A102769, A131595, A179290, A179292, A179294.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A187110 (tetrahedron). - Stanislav Sykora, Feb 10 2014

RealDigits[(Sqrt[3]+Sqrt[15])/4, 10, 175][[1]]

(1+sqrt(5))*sqrt(3)/4 \\ Stefano Spezia, Jan 27 2025

Equals (sqrt(3) + sqrt(15))/4 = sqrt((9 + 3*sqrt(5))/8).
The minimal polynomial is 16*x^4 - 36*x^2 + 9. - Joerg Arndt, Feb 05 2014
Equals (sqrt(3)/2) * phi = A010527 * A001622. - Amiram Eldar, Jun 02 2023

A268682 Decimal expansion of 1 - 1/sqrt(2).

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Feb 19 2016

This is the maximum fraction of mass-energy of a black hole which can come from angular momentum, and hence the maximum energy which can be extracted from the black hole via the Penrose process.
Differs from A157215 only in one or two leading digits. - R. J. Mathar, Feb 24 2016
This is the probability that a randomly selected vertex in a random Schroeder tree is a leaf as the number of leaves goes to infinity. See Corollary 2.1.2. of Van Duzer. - Michel Marcus, Apr 12 2019

			0.29289321881345247559915563789515096071516406231152596341166013100463376...

Charles D. Dermer and Govind Menon, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos (2009). See pp. 400-402.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.
Stijn J. van Tongeren, Rotating black holes (2009).
Wikipedia, Penrose process
Index entries for algebraic numbers, degree 2

Cf. A010503, A268683.

1-1/Sqrt(2); // Vincenzo Librandi, Feb 20 2016

RealDigits[1 - 1 / Sqrt[2], 10, 90] [[1]] (* Vincenzo Librandi, Feb 20 2016 *)

PARI
```
1-sqrt(.5)
```

Equals 1 - A010503.
a(n) = 9 - A010503(n). - Philippe Deléham, Feb 21 2016
Equals Integral_{x=0..Pi/4} sin(x) dx. - Amiram Eldar, Jun 29 2020

More digits from Jon E. Schoenfield, Mar 15 2018

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)

A073084 Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 17 2002

The equation has three solutions, x = 2, 4 and -0.76666469596....
-x is the power tower (tetration) of 1/sqrt(2) (A010503), also equal to LambertW(log(sqrt(2)))/log(sqrt(2)). - Stanislav Sykora, Nov 04 2013
x is transcendental by the Gelfond-Schneider theorem. Proof: If we accept that x is not an integer, then we can see that x is not rational. For if it were, x^2 would be as well, whereas 2^x would not be (because 2 is not a perfect power). Thus we would have a contradiction (since x^2 = 2^x). Similarly, if x were irrational algebraic, x^2 would be as well, while 2^x would be transcendental (by the Gelfond-Schneider theorem). Thus the only conclusion is that x is transcendental. - Chayim Lowen, Aug 13 2015
From Robert G. Wilson v, May 18 2021: (Start)
Let W be the Lambert power log function,
f(x) = e^(-W_x(-log(sqrt(2)))) and g(x) = -e^(-W_x(log(sqrt(2)))).
Then f(0)=2, f(-1)= 4 and g(0) = c. Except for these three illustrated examples, all integer arguments x yield a complex solution which satisfies the equation. (End)
x is also the negative solution to 4^x = x^4, which reduces to 2^x = x^2 upon taking the square root of both sides. - Jason Bard, Aug 16 2025

			0.76666469596212309311120442251031484800...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.11, p. 449.

Stanislav Sykora, Table of n, a(n) for n = 0..1999
RJMilazzo and others, largest solution to 2^x=x^2, thread in newsgroup sci.math, Aug 17, 2002.
Eric Weisstein's World of Mathematics, Power.
Index entries for transcendental numbers.

Cf. A010503, A344905.

Cf. non-integer solutions to a^x = x^a: A166928 (a = 3), A341328 (a = 5).

evalf((f-> LambertW(f)/f)(log(2)/2), 145);  # Alois P. Heinz, Aug 03 2023

RealDigits[NSolve[2^x == x^2, x, WorkingPrecision -> 150][[1, 1]][[2]]][[1]]
c = -Exp[-LambertW[Log[2]/2]]; RealDigits[c, 10, 111][[1]] (* Robert G. Wilson v, May 18 2021 *)
(* To view the two curves: *) Plot[{2^x, x^2}, {x, -4.5, 4.5}] (* Robert G. Wilson v, May 18 2021 *)
RealDigits[-x/.FindRoot[2^x==x^2,{x,-1},WorkingPrecision->120],10,120][[1]] (* Harvey P. Dale, Jul 15 2023 *)

lambertw(log(sqrt(2)))/log(sqrt(2)) \\ Stanislav Sykora, Nov 04 2013

-2*LambertW(log(2)/2)/log(2). - Eric W. Weisstein, Jan 23 2005

Equals 1/A344905. - Hugo Pfoertner, Dec 18 2024

Offset corrected by R. J. Mathar, Feb 05 2009

A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, Nov 29 2013

The corresponding real part is in A232735.
Root of the equation 1 - 4*x - 4*x^2 + 8*x^3 = 0. - Vaclav Kotesovec, Apr 04 2021
The other 2 roots are -A362922 and A073052. - R. J. Mathar, Aug 29 2025

			0.222520933956314404288902564496794759466355568764544955311987...

Stanislav Sykora, Table of n, a(n) for n = 0..1000
Index entries for algebraic numbers, degree 3.

Cf. A232735 (real part), A010503 (imag(I^(1/2))), A182168 (imag(I^(1/4))), A019827 (imag(I^(1/5))), A019824 (imag(I^(1/6))), A232738 (imag(I^(1/8))), A019819 (imag(I^(1/9))), A019818 (imag(I^(1/10))).

A023365 a(n) = 2^(3^(n-1)).

Original entry on oeis.org

2, 8, 512, 134217728, 2417851639229258349412352, 14134776518227074636666380005943348126619871175004951664972849610340958208

Offset: 1

David W. Wilson

nonn

a(n+1) = a(n) converted to base 8 from base 2 (written in base 10).
Number of disjunctive-normal forms of n-1 variables (either with x, or x-negated or without x). - Labos Elemer, Jul 24 2003
a(n)*Psi(3^n,x), with the (monic) minimal polynomial Psi(3^n,x) of cos(2*Pi/3^n), becomes an integer polynomial with coefficient 1 of x^0.
  E.g., 8*Psi(9,x)=8*(x^3 - (3/4)*x + 1/8) = 8*x^3 - 6*x + 1.
  See A181875/A181876, A181877 and the W. Lang link under A181875. - Wolfdieter Lang, Feb 24 2011
The next term (a(7)) has 220 digits. - Harvey P. Dale, Aug 10 2014
These seem to be the reduced denominators of Newton's iteration for 1/sqrt(2), starting with 1/2. - Steven Finch, Oct 08 2024

Vincenzo Librandi, Table of n, a(n) for n = 1..8
W. van der Aalst, J. Buijs and B. van Dongen, Towards Improving the Representational Bias of Process Mining, 2012. - From _N. J. A. Sloane_, Feb 03 2013
X. Gourdon and P. Sebah, Pythagoras' Constant. - From _Steven Finch_, Oct 07 2024

Cf. A000079, A000244, A010503, A181875, A181876, A181877, A376867.

[Floor(2^(3^(n-1))): n in [1..10]]; // Vincenzo Librandi, Aug 11 2014

NestList[#^3&,2,6] (* Harvey P. Dale, Aug 10 2014 *)

def A023365(n): return 1<<3**(n-1) # Chai Wah Wu, Oct 11 2024

[2^(3^(n-1)) for n in range(1,8)] # Stefano Spezia, Oct 15 2024

a(n) = a(n-1)^3.
a(n) = A000079(A000244(n-1)).
a(n+1) is conjectured to be the reduced denominator of b(n) = b(n-1)*(3/2 - b(n-1)^2); b(0) = 1/2. - Steven Finch, Oct 08 2024
Limit_{n -> oo} A376867(n-1)/a(n) = 1/sqrt(2) = A010503. - Steven Finch, Oct 08 2024

A187110 Decimal expansion of sqrt(3/8).

Original entry on oeis.org

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

Offset: 0

Vladimir Joseph Stephan Orlovsky, Mar 05 2011

Apart from leading digits, the same as A174925.

Radius of the circumscribed sphere (congruent with vertices) for a regular tetrahedron with unit edges. - Stanislav Sykora, Nov 20 2013

			sqrt(3/8)=0.61237243569579452454932101867647284799148687016417..

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

Michael Penn, Maximize the area of one ellipse!, YouTube video, 2020.
Wikipedia, Tetrahedron.
Index entries for algebraic numbers, degree 2.

Cf. A002194, A010464, A010466, A020781, A040001, A040005, A115754, A301755.

Cf. Platonic solids circumradii: A010503 (octahedron), A010527 (cube), A019881 (icosahedron), A179296 (dodecahedron). - Stanislav Sykora, Feb 10 2014

Mathematica
```
RealDigits[N[Sqrt[3/8],300]][[1]]
```

sqrt(3/8) \\ Charles R Greathouse IV, Mar 16 2022

Equals A010464/4. - Stefano Spezia, Jan 26 2025

Equals 3*A020781 = A115754/2 = sqrt(A301755). - Hugo Pfoertner, Jan 26 2025

A040042 Continued fraction for sqrt(50) = 5*sqrt(2).

Original entry on oeis.org

7, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14

Offset: 0

N. J. A. Sloane

			7.07106781186547524400844... = 7 + 1/(14 + 1/(14 + 1/(14 + 1/(14 + ...)))). - _Harry J. Smith_, Jun 01 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 (1).

Cf. A010503 (decimal expansion), A041084/A041085 (convergents), A248275 (Egyptian fraction).

Cf. A040000.

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

ContinuedFraction[Sqrt[50],300] (* Vladimir Joseph Stephan Orlovsky, Mar 07 2011 *)

{ allocatemem(932245000); default(realprecision, 47000); x=contfrac(sqrt(50)); for (n=0, 20000, write("b040042.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 01 2009

From Elmo R. Oliveira, Feb 07 2024: (Start)
a(n) = 14 = A010853(n) for n >= 1.
G.f.: 7*(1+x)/(1-x).
E.g.f.: 14*exp(x) - 7.
a(n) = 7*A040000(n). (End)

A157214 Decimal expansion of 18 + 5*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			18+5*sqrt(2) = 25.07106781186547524400...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Index entries for algebraic numbers, degree 2

Cf. A129544, A157213, A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18-5*sqrt(2))/(18+5*sqrt(2))).

[18 + 5*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[18+5Sqrt[2],10,150][[1]]  (* Harvey P. Dale, Mar 18 2011 *)

18 + 5*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 18 + 10*A010503. - R. J. Mathar, Feb 27 2009

cp's OEIS Frontend

A099972 Write 1/sqrt(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A179296 Decimal expansion of circumradius of a regular dodecahedron with edge length 1.

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A268682 Decimal expansion of 1 - 1/sqrt(2).

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A020763 Decimal expansion of 1/sqrt(6).

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A073084 Decimal expansion of -x, where x is the negative solution to the equation 2^x = x^2.

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A232736 Decimal expansion of sin(Pi/14), or the imaginary part of (-1)^(1/7).

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