A002194 -id:A002194 - cp's OEIS frontend

A165952 Decimal expansion of 2sqrt(3)/(3Pi).

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

Offset: 0

Rick L. Shepherd, Oct 02 2009

The ratio of the volume of a cube to the volume of the circumscribed sphere (which has circumradius a*sqrt(3)/2 = a*A010527, where a is the cube's edge length; see MathWorld link). For similar ratios for other Platonic solids, see A165922, A049541, A165953, and A165954. A063723 shows the order of these by size.

			0.3675525969478613663408843322086462942649243202444271018662440135273535356...

Eric Weisstein's World of Mathematics, Cube.
Index entries for transcendental numbers

Cf. A000796, A002194, A165922, A049541, A165953, A165954, A063723, A020760, A020832.

RealDigits[(2*Sqrt[3])/(3Pi),10,120][[1]] (* Harvey P. Dale, Oct 08 2012 *)

PARI
```
2*sqrt(3)/(3*Pi)
```

2*sqrt(3)/(3*Pi) = 2*A002194/(3*A000796) = 3*A165922 = (2*sqrt(3)/3)*A049541 = 10*A020832*A049541 = 2*A020760*A049541.

A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

Original entry on oeis.org

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

Offset: 0

Stanislav Sykora, May 02 2016

15-gon is the first m-gon with odd composite m which is constructible (see A003401) in virtue of the fact that 15 is the product of two distinct Fermat primes (A019434). The next such case is 51-gon (m=3*17), followed by 85-gon (m=5*17), 771-gon (m=3*257), etc.
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 4, pp. 69-74. See also the comments in A302711 with a link to Romanus' book, Exemplum quaesitum.
This problem is equivalent to R(45, 2*sin(Pi/675)) = 2*sin(Pi/15), with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/675) see A302716. (End)

			0.415823381635518674203484568810250332433169521255447672814363947...

Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.

Stanislav Sykora, Table of n, a(n) for n = 0..2000
Mauro Fiorentini, Construibili (numeri)
Eric Weisstein's World of Mathematics, Constructible Number
Wikipedia, Constructible number
Wikipedia, Regular polygon
Index entries for sequences related to Chebyshev polynomials.
Index entries for algebraic numbers, degree 8.

Cf. A003401, A019434, A127672, A302711, A302716.

Edge lengths of other constructible m-gons: A002194 (m=3), A002193 (4), A182007 (5), A101464 (8), A094214 (10), A101263 (12), A272535 (16), A228787 (17), A272536 (20).

RealDigits[N[2Sin[Pi/15], 100]][[1]] (* Robert Price, May 02 2016*)

PARI
```
2*sin(Pi/15)
```

Equals 2*sin(Pi/m) for m=15, 2*A019821.
Also equals (sqrt(3) - sqrt(15) + sqrt(10 + 2*sqrt(5)))/4.
Also equals sqrt(7 - sqrt(5) - sqrt(30 - 6*sqrt(5)))/2. This is the rewritten expression of the Havil reference on top of p. 70. - Wolfdieter Lang, Apr 29 2018

A377602 Decimal expansion of the surface area of a snub cube (snub cuboctahedron) with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 02 2024

			19.856406460551018348219570732046978935542442030...

Eric Weisstein's World of Mathematics, Snub Cube.
Wikipedia, Snub cube.
Index entries for algebraic numbers, degree 2.

Cf. A377603 (volume), A377604 (circumradius), A377605 (midradius).

Cf. A002194, A200243.

First[RealDigits[6 + Sqrt[192], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubCube", "SurfaceArea"], 10, 100]]

Equals 6 + 8*sqrt(3) = 6 + 8*A002194 = 6 + A200243.

A377804 Decimal expansion of the surface area of a snub dodecahedron with unit edge length.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, Nov 08 2024

			55.2867449584451489436570705587807625317445951163...

Eric Weisstein's World of Mathematics, Snub Dodecahedron.
Wikipedia, Snub dodecahedron.
Index entries for algebraic numbers, degree 8.

Cf. A377805 (volume), A377806 (circumradius), A377807 (midradius).
Cf. A131595 (analogous for a regular dodecahedron).
Cf. A002194.

First[RealDigits[20*Sqrt[3] + 3*Sqrt[25 + 10*Sqrt[5]], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["SnubDodecahedron", "SurfaceArea"], 10, 100]]

Equals 20*sqrt(3) + 3*sqrt(25 + 10*sqrt(5)) = 20*A002194 + A131595.

A384215 Decimal expansion of the surface area of a gyroelongated square cupola with unit edge.

Original entry on oeis.org

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

Offset: 2

Paolo Xausa, May 23 2025

The gyroelongated square cupola is Johnson solid J_23.

			18.4886811625905765652406091559487579918533700198...

Paolo Xausa, Table of n, a(n) for n = 2..10000
Wikipedia, Gyroelongated square cupola.

Cf. A384214 (volume).

Cf. A002194, A010466.

First[RealDigits[7 + Sqrt[8] + Sqrt[75], 10, 100]] (* or *)
First[RealDigits[PolyhedronData["J23", "SurfaceArea"], 10, 100]]

Equals 7 + 2*sqrt(2) + 5*sqrt(3) = 7 + A010466 + 5*A002194.

Equals the largest root of x^4 - 28*x^3 + 128*x^2 + 952*x - 1244.

A010538 Decimal expansion of square root of 87.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 9 followed by {3, 18} repeated. - Harry J. Smith, Jun 10 2009

			9.3273790530888150455544755423205569832762406...

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

Cf. A040077 (continued fraction).

Cf. A002194, A010484.

RealDigits[N[87^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)

from math import isqrt
def aupton(nn): return list(map(int, str(isqrt(87 * 10**(2*nn)))))[:nn]
print(aupton(100)) # Michael S. Branicky, Sep 03 2021

Equals A002194 * A010484. - R. J. Mathar, Jun 08 2020

A119343 Theodorus primes: primes formed from the concatenation of the initial decimal digits of Theodorus's constant, sqrt(3).

Original entry on oeis.org

17, 173, 1732050807568877293

Offset: 1

Eric W. Weisstein, May 15 2006

The next term (a(4)) has 111 digits. - Harvey P. Dale, Oct 24 2014

			sqrt(3) = 1.732050807568877..., 17, the concatenation of the first 2 decimal digits, is prime, so a(1) = 17.

Amiram Eldar, Table of n, a(n) for n = 1..6
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A002194, A119344.

nn=300;With[{ds3=RealDigits[Sqrt[3],10,nn][[1]]},Select[ Table[ FromDigits[ Take[ds3,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Oct 24 2014 *)

Edited by Charles R Greathouse IV, Apr 27 2010

A153594 a(n) = ((4 + sqrt(3))^n - (4 - sqrt(3))^n)/(2*sqrt(3)).

Original entry on oeis.org

1, 8, 51, 304, 1769, 10200, 58603, 336224, 1927953, 11052712, 63358307, 363181200, 2081791609, 11932977272, 68400527259, 392075513536, 2247397253921, 12882196355400, 73841406542227, 423262699717616, 2426163312691977, 13906891405206808

Offset: 1

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Second binomial transform of A054491. Fourth binomial transform of 1 followed by A162766 and of A074324 without initial term 1.
First differences are in A161728.
Lim_{n -> infinity} a(n)/a(n-1) = 4 + sqrt(3) = 5.73205080756887729....

G. C. Greubel, Table of n, a(n) for n = 1..500
Index entries for linear recurrences with constant coefficients, signature (8,-13).

Cf. A002194 (decimal expansion of sqrt(3)), A054491, A074324, A161728, A162766.

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-3); S:=[ ((4+r)^n-(4-r)^n)/(2*r): n in [1..21] ]; [ Integers()!S[j]: j in [1..#S] ];  // Klaus Brockhaus, Dec 31 2008

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1)-13*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Join[{a=1,b=8},Table[c=8*b-13*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-13},{1,8},40] (* Harvey P. Dale, Aug 16 2012 *)

a(n)=([0,1; -13,8]^(n-1)*[1;8])[1,1] \\ Charles R Greathouse IV, Sep 04 2016

[lucas_number1(n,8,13) for n in range(1, 22)] # Zerinvary Lajos, Apr 23 2009

G.f.: x/(1 - 8*x + 13*x^2). - Klaus Brockhaus, Dec 31 2008, corrected Oct 11 2009
a(n) = 8*a(n-1) - 13*a(n-2) for n > 1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009
E.g.f.: sinh(sqrt(3)*x)*exp(4*x)/sqrt(3). - Ilya Gutkovskiy, Aug 23 2016
a(n) = Sum_{k=0..n-1} A027907(n,2k+1)*3^k. - J. Conrad, Aug 30 2016
a(n) = Sum_{k=0..n-1} A083882(n-1-k)*4^k. - J. Conrad, Sep 03 2016

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

A196530 Decimal expansion of log(2+sqrt(3))/sqrt(3).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=1.

			0.7603459963009463475310942548...

L. B. W. Jolley, Summation of series, Dover (1961), eq. (83), page 16.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
E. D. Krupnikov, K. S. Kolbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95
R. J. Mathar, Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table in section 2.2, L(m=12,r=4,s=1).
Index entries for transcendental numbers.

Cf. A065918, A002194, A110161.

RealDigits[Log[2 + Sqrt[3]]/Sqrt[3], 10, 89][[1]] (* Bruno Berselli, Dec 20 2011 *)

log(sqrt(3)+2)/sqrt(3) \\ Charles R Greathouse IV, May 15 2019

Equals A065918/A002194.
Equals Sum_{n>=1} A110161(n)/n.
Equals Sum_{k>=1} (-1)^(k+1)*2^k/(k * binomial(2*k,k)). - Amiram Eldar, Aug 19 2020
Equals 1/Product_{p prime} (1 - Kronecker(12,p)/p), where Kronecker(12,p) = 0 if p = 2 or 3, 1 if p == 1 or 11 (mod 12) or -1 if p == 5 or 7 (mod 12). - Amiram Eldar, Dec 17 2023
Equals A259830 - 2. - Hugo Pfoertner, Apr 06 2024
Equals (1/2)*2F1(1/2,1;3/2;3/4) [Krupnikov] - R. J. Mathar, Jun 11 2024

A240935 Decimal expansion of 3sqrt(3)/(4Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Aug 03 2014

A triangle of maximal area inside a circle is necessarily an inscribed equilateral triangle. This constant is the ratio of the triangle's area to the circle's area. In general, the ratio of an arbitrary triangle's area to the area of its unique Steiner ellipse, which has the least area of any circumscribed ellipse (an equilateral triangle's Steiner ellipse is a circle).

Also the probability that the distance between 2 randomly selected points within a circle will be larger than the radius. - Amiram Eldar, Mar 03 2019

			0.4134966715663440371334948737347270810480...

B. F. Finkel, Problem 38, solved by O. W. Anthony, Henry Heaton, and G. B. M. Zerr, The American Mathematical Monthly, Vol. 3, No. 12 (1896), pp. 324-326.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), p.52
I. Todhunter, A treatise on the Integral Calculus, London and Cambridge: MacMillan and Co., 1868, page 320, Example 7.
Wikipedia, Steiner ellipse
Index entries for transcendental numbers

Cf. A073010, A060294, A003881, A002194, A000796, A102519, A086089.

Digits:=100: evalf(3*sqrt(3)/(4*Pi)); # Wesley Ivan Hurt, Aug 03 2014

Flatten[RealDigits[3 Sqrt[3]/(4 Pi), 10, 100, -1]] (* Wesley Ivan Hurt, Aug 03 2014 *)

default(realprecision, 120);
3*sqrt(3)/(4*Pi)

3*sqrt(3)/(4*Pi) = 3*A002194/(4*A000796).

Equals A093604^2. - Hugo Pfoertner, May 18 2024

cp's OEIS Frontend

A165952 Decimal expansion of 2*sqrt(3)/(3*Pi).

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A272534 Decimal expansion of the edge length of a regular 15-gon with unit circumradius.

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A377602 Decimal expansion of the surface area of a snub cube (snub cuboctahedron) with unit edge length.

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A377804 Decimal expansion of the surface area of a snub dodecahedron with unit edge length.

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A384215 Decimal expansion of the surface area of a gyroelongated square cupola with unit edge.

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A010538 Decimal expansion of square root of 87.

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A119343 Theodorus primes: primes formed from the concatenation of the initial decimal digits of Theodorus's constant, sqrt(3).

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A165952 Decimal expansion of 2sqrt(3)/(3Pi).

A240935 Decimal expansion of 3sqrt(3)/(4Pi).