A073005 -id:A073005 - cp's OEIS frontend

A002194 Decimal expansion of sqrt(3).

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

Offset: 1

N. J. A. Sloane

"The square root of 3, the 2nd number, after root 2, to be proved irrational, by Theodorus."
Length of a diagonal between any vertex of the unit cube and the one corresponding (opposite) vertex not part of the three faces meeting at the original vertex. (Diagonal is hypotenuse of a triangle with sides 1 and sqrt(2)). Hence the diameter of the sphere circumscribed around the unit cube; the ratio of the diameter of any sphere to the edge length of its inscribed cube. - Rick L. Shepherd, Jun 09 2005
The square root of 3 is the length of the minimal Y-shaped (symmetrical) network linking three points unit distance apart. - Lekraj Beedassy, Apr 12 2006
Continued fraction expansion is 1 followed by {1, 2} repeated. - Harry J. Smith, Jun 01 2009
Also, tan(Pi/3) = 2 sin(Pi/3). - M. F. Hasler, Oct 27 2011
Surface of regular tetrahedron with unit edge. - Stanislav Sykora, May 31 2012
This is the case n=6 of Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)) = 2*cos(Pi/n), therefore sqrt(3) = A175379*A203145/(A073005*A073006). - Bruno Berselli, Dec 13 2012
Ratio of base length to leg length in the isosceles "vampire" triangle, that is, the only isosceles triangle without reflection triangle. The product of cosines of the internal angles of a triangle with sides 1, 1 and sqrt(3) and all similar triangles is -3/8. Hence its reflection triangle is degenerate. See the link below. - Martin Janecke, May 09 2013
Half of the surface of regular octahedron with unit edge (A010469), and one fifth that of a regular icosahedron with unit edge (i.e., 2*A010527). - Stanislav Sykora, Nov 30 2013
Diameter of a sphere whose surface area equals 3*Pi. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
Sometimes called Theodorus's constant, after the ancient Greek mathematician Theodorus of Cyrene (5th century BC). - Amiram Eldar, Apr 02 2022
For any triangle ABC, cotan(A) + cotan(B) + cotan(C) >= sqrt(3); equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - Bernard Schott, Sep 13 2022

			1.73205080756887729352744634150587236694280525381038062805580697945193...

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 184.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.4 Irrational Numbers and §12.4 Theorems and Formulas (Solid Geometry), pp. 84, 450.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Revised Edition, Penguin Books, London, England, 1997, page 23.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Madeleine Bonsma-Fisher and Kent Bonsma-Fisher, How big a table do you need for your jigsaw puzzle?, arXiv:2312.04588 [math.HO], 2023.
M. F. Jones, 22900D approximations to the square roots of the primes less than 100, Math. Comp., Vol. 22, No. 101 (1968), pp. 234-235.
Kiran S. Kedlaya, A < B, (1999) Problem 6.4, p. 6.
Jason Kimberley, Index of expansions of sqrt(d) in base b.
Robert J. Nemiroff and Jerry Bonnell, The first 1 million digits of the square root of 3.
Simon Plouffe, Plouffe's Inverter, The square root of 3 to 10 million digits.
Horace S. Uhler, Approximations exceeding 1300 decimals for sqrt 3, 1/sqrt 3, sin(pi/3) and distribution of digits in them, Proc. Nat. Acad. Sci. U. S. A., Vol. 37, No. 7 (1951), pp. 443-447.
Eric Weisstein's World of Mathematics, Reflection Triangle.
Eric Weisstein's World of Mathematics, Square Root.
Eric Weisstein's World of Mathematics, Theodorus's Constant.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 2

Cf. A040001 (continued fraction), A220335.

Cf. A010469 (double), A010527 (half), A131595 (surface of regular dodecahedron).

SetDefaultRealField(RealField(100)); Sqrt(3); // G. C. Greubel, Aug 21 2018

evalf(sqrt(3), 100); # Michal Paulovic, Feb 24 2023

Mathematica
```
RealDigits[Sqrt[3], 10, 100][[1]]
```

default(realprecision, 20080); x=(sqrt(3)); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002194.txt", n, " ", d));  \\ Harry J. Smith, Jun 01 2009

Equals Sum_{k>=0} binomial(2*k,k)/6^k = Sum_{k>=0} binomial(2*k,k) * k/6^k. - Amiram Eldar, Aug 03 2020
sqrt(3) = 1 + 1/2 + 1/(2*3) + 1/(2*3*4) + 1/(2*3*4*2) + 1/(2*3*4*2*8) + 1/(2*3*4*2*8*14) + 1/(2*3*4*2*8*14*2) + 1/(2*3*4*2*8*14*2*98) + 1/(2*3*4*2*8*14*2*98*194) + .... (Define F(n) = (n-1)*sqrt(n^2 - 1) - (n^2 - n - 1). Show F(n) = 1/2 + 1/(2*(n+1)) +  1/(2*(n+1)*(2*n)) + 1/(2*(n+1)*(2*n))*F(2*n^2 - 1) for n >= 0; then iterate this identity at n = 2. See A220335.) - Peter Bala, Mar 18 2022
Equals i^(1/3) + i^(-1/3). - Gary W. Adamson, Jul 06 2022
Equals Product_{n>=1} 3^(1/3^n). - Michal Paulovic, Feb 24 2023
Equals Product_{n>=0} ((6*n + 2)*(6*n + 4))/((6*n + 1)*(6*n + 5)). - Antonio Graciá Llorente, Feb 22 2024
Equals tan(Pi/3) = A010527/(1/2). - R. J. Mathar, Aug 31 2025

More terms from Robert G. Wilson v, Dec 07 2000

A002161 Decimal expansion of square root of Pi.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also Gamma(1/2). - Franklin T. Adams-Watters, Apr 07 2006
The integral of the Gaussian function exp(-x^2) over the real line. - Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008
Also equals the average distance between two points in two dimensions where coordinates are independent normally distributed random variables with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014, after Steven Finch
Also diameter of a sphere whose surface area equals Pi^2. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
Convergents of continued fractions: 7/4, 16/9, 23/13, 39/22, 257/145, 296/167, 8545/4821, ... - R. J. Mathar, Jan 29 2025

			1.7724538509055160272981674833411451827975494561223871282138...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 190.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, page 413.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Donald Knuth, Why pi?, Christmas Tree lecture, Dec 06 2010 (video).
Index entries for sequences related to the number Pi.
Index entries for transcendental numbers.

Cf. A000796, A002388, A073006, A195907, A203145.

Cf. decimal expansions of Gamma(1/k): A073005 (k=3), A068466 (k=4), A175380 (k=5), A175379 (k=6), A220086 (k=7), A203142 (k=8).

R:= RealField(100); Sqrt(Pi(R));  // G. C. Greubel, Mar 10 2018

evalf(sqrt(Pi),120); # Muniru A Asiru, Nov 11 2018

RealDigits[N[Sqrt[Pi], 120]][[1]] (* Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008 *)

default(realprecision, 20080); x=sqrt(Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002161.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009

Equals (1/2) * Sum_{n>=0} ((-1)^n * (4*n+1) * (1/8)^(n+1) * (2^(n+1))^3 * Gamma(n+1/2)^3 / Gamma(n+1)^3). - Alexander R. Povolotsky, Mar 25 2013
Equals Integral_{x=0..1} 1/sqrt(-log(x)) dx. - Jean-François Alcover, Apr 29 2013
Equals Sum_{k>=0} (k+1/2)!/(k+2)!. - Amiram Eldar, Jun 19 2023
Equals Integral_{x=0..oo} exp(-x)/sqrt(x) dx. - Michal Paulovic, Sep 24 2023
Equals Integral_{x=0..oo} 4/(exp(x^2)*(2*x^2 + 1)^2) dx. - Kritsada Moomuang, Jun 05 2025

More terms from Franklin T. Adams-Watters, Apr 07 2006

A073006 Decimal expansion of Gamma(2/3).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

This constant is transcendental: Chudnovsky famously proved that Gamma(1/3) is algebraically independent of Pi, but Gamma(1/3)*Gamma(2/3) = 2*Pi/sqrt(3) by the reflection formula. - Charles R Greathouse IV, Aug 21 2023

			1.354117939426400416945288028154513785519327266056793698394022467963782...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Simon Plouffe, GAMMA(2/3).
Index to sequences related to the Gamma function.
Index entries for transcendental numbers.

Cf. A030652 (continued fraction). - Harry J. Smith, May 14 2009

Cf. A073005, A186706.

SetDefaultRealField(RealField(100)); Gamma(2/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[2/3], 110]][[1]]
```

allocatemem(932245000); default(realprecision, 5080); x=gamma(2/3); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b073006.txt", n, " ", d));  \\ Harry J. Smith, May 14 2009

Gamma(2/3) * A073005 = A186706. - R. J. Mathar, Jun 18 2006

A047657 Sextuple factorial numbers: a(n) = Product_{k=0..n-1} (6*k+2).

Original entry on oeis.org

1, 2, 16, 224, 4480, 116480, 3727360, 141639680, 6232145920, 311607296000, 17450008576000, 1081900531712000, 73569236156416000, 5444123475574784000, 435529878045982720000, 37455569511954513920000, 3445912395099815280640000, 337699414719781897502720000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

G. C. Greubel, Table of n, a(n) for n = 0..340

Cf. A007559, A008542, A011781, A073005.

Cf. A000165, A008544, A001813, A047055, A084947, A084948, A084949.

List([0..20], n-> Product([0..n-1], k-> 6*k+2) ); # G. C. Greubel, Aug 18 2019

[1] cat [(&*[6*k+2: k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 18 2019

a:= n->product(6*j+2, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 18 2019

b[1]=2; b[n_]:= b[n] = b[n-1] +6; a[0]=1; a[1]=2; a[n_]:= a[n] = a[n-1]*b[n]; Table[a[n], {n,0,20}] (* Roger L. Bagula, Sep 17 2008 *)
FoldList[Times,1,6*Range[0,20]+2] (* Harvey P. Dale, Aug 06 2013 *)
Table[6^n*Pochhammer[1/3, n], {n,0,20}] (* G. C. Greubel, Aug 18 2019 *)

vector(20, n, n--; prod(k=0, n-1, 6*k+2)) \\ G. C. Greubel, Aug 18 2019

[product(6*k+2 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 18 2019

E.g.f.: (1-6*x)^(-1/3).
a(n) = 2^n*A007559(n).
a(n) = A084941(n)/A000142(n)*A000079(n) = 6^n*Pochhammer(1/3, n) = 1/2*6^n*Gamma(n+1/3)*sqrt(3)*Gamma(2/3)/Pi. - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
Let b(n) = b(n-1) + 6; then a(n) = b(n)*a(n-1). - Roger L. Bagula, Sep 17 2008
G.f.: 1/(1-2*x/(1-6*x/(1-8*x/(1-12*x/(1-14*x/(1-18*x/(1-20*x/(1-24*x/(1-26*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^n*Sum_{k=0..n} (3/2)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/G(0) where G(k) = 1 - x*(6*k+2)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
D-finite with recurrence: a(n) +2*(-3*n+2)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/6)*(Gamma(1/3) - Gamma(1/3, 1/6))/6^(2/3). - Amiram Eldar, Dec 18 2022
a(n) ~ sqrt(Pi) * 2^(n+1/2) * (3/e)^n * n^(n-1/6) / Gamma(1/3). - Amiram Eldar, Sep 01 2025

A084941 Octagorials: n-th polygorial for k=8.

Original entry on oeis.org

1, 1, 8, 168, 6720, 436800, 41932800, 5577062400, 981562982400, 220851671040000, 61838467891200000, 21086917550899200000, 8603462360766873600000, 4138265395528866201600000, 2317428621496165072896000000, 1494741460865026472017920000000, 1100129715196659483405189120000000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A073005, A084939, A084940, A084942, A084943, A084944, A085356.

a := n->n!/2^n*product(6*i+2,i=0..n-1); [seq(a(j),j=0..30)];

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[8, #] &, 16, 0] (* Robert G. Wilson v, Dec 26 2016 *)

a(n) = n! / 2^n * prod(i=0, n-1, 6*i+2) \\ Felix Fröhlich, Dec 13 2016

a(n) = polygorial(n, 8) = (A000142(n)/A000079(n))*A047657(n) = (n!/2^n)*Product_{i=0..n-1} (6*i+2) = (n!/2^n)*6^n*Pochhammer(1/3, n) = (n!/2)*3^n*sqrt(3)*Gamma(n+1/3)*Gamma(2/3)/Pi.
D-finite with recurrence a(n) = n*(3*n-2)*a(n-1). - R. J. Mathar, Mar 12 2019
a(n) ~ 2 * 3^n * n^(2*n + 1/3) * Pi /(Gamma(1/3) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A175379 Decimal expansion of Gamma(1/6).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Apr 24 2010

A175379 * A073005 * A002161 * A073006 * A203145 = 4*sqrt(Pi^5/3), which is the case n=6 of Product_{i=1..n-1} Gamma(i/n) = sqrt((2*Pi)^(n-1)/n). - Bruno Berselli, Dec 18 2012

The transcendence of this constant is in the mathematical folklore; see Finch (who credits Nesterenko) and Gun-Murty-Rath. - Charles R Greathouse IV, Nov 11 2013

			Equals 5.56631600178023...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven Finch, Errata and addenda to Mathematical Constants (2013)
Sanoli Gun, M. Ram Murty, and Purusottam Rath, Transcendence of the log gamma function and some discrete periods, J. Number Theory (2009), doi:10.1016/j.jnt.2009.01.008
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Wikipedia, Particular values of the Gamma function: General rational arguments
Index entries for transcendental numbers
Index entries to sequences related to the Gamma function

Cf. A073006, A203145.

SetDefaultRealField(RealField(100)); Gamma(1/6); // G. C. Greubel, Mar 10 2018

Maple
```
evalf(GAMMA(1/6)) ;
```

RealDigits[Gamma[1/6], 10, 110][[1]] (* Bruno Berselli, Dec 13 2012 *)

gamma(1/6) \\ Charles R Greathouse IV, Nov 16 2013

Equals 2*Pi/A203145 = A002194 * A073005^2 / (A002161 * A002580) = A019692 / 1.12878703....

A186706 Decimal expansion of the Integral of Dedekind Eta(x*I) from x = 0..oo.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Feb 25 2011

Reduction of the integral by Robert Israel, Jul 25 2012: (Start)
Use the definition of DedekindEta as a sum:
Eta(i*x) = Sum_{n=-oo..oo} (-1)^n*exp(-Pi*x*(6n-1)^2/12).
Now Integral_{x=0..oo} exp(-Pi*x*(6n-1)^2/12) dx = 12/(Pi*(6n-1)^2).
According to Maple, Sum_{n=-oo..oo} (-1)^n*12/(Pi*(6n-1)^2) is
2*3^(1/2)*(dilog(1-(1/2)*i-(1/2)*3^(1/2)) - dilog(1-(1/2)*i+(1/2)*3^(1/2)) - dilog(1+(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)))/Pi
(Jonquiere's inversion formula -- see http://en.wikipedia.org/wiki/Polylogarithm)
(but note that Maple's dilog(z) is L_2(1-z) in the notation there) gives
dilog(1-(1/2)*i-(1/2)*3^(1/2)) + dilog(1+(1/2)*i-(1/2)*3^(1/2)) = (13/72)*Pi^2
and
dilog(1-(1/2)*i+(1/2)*3^(1/2)) + dilog(1+(1/2)*i+(1/2)*3^(1/2)) = -11*Pi^2/72
which give the desired multiple of Pi. (End)
Ratio of surface area of a sphere to the regular octahedron whose edge equals the radius of the sphere. - Omar E. Pol, Dec 30 2023

			3.627598728468435701188156515284311464568132496185481151139769870776...

Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 149.

D. H. Lehmer, Interesting series involving the central binomial coefficient, Am. Math. Monthly 92 (7) (1985) 449.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 511.
Eric W. Weisstein's World of Mathematics, Dedekind Eta Function.

Cf. A073005, A073006, A093602, A093766,

RealDigits[2 Pi/Sqrt[3], 10, 111][[1]] (* Robert G. Wilson v, Nov 18 2012 *)

intnum(x=1e-7,1e6,eta(x*I,1)) \\ Charles R Greathouse IV, Feb 26 2011

Equals 2*Pi/sqrt(3), 2 times A093602, and in consequence equal to Sum_{m>=1} 3^m/(m*binomial(2m,m)) according to Lehmer. - R. J. Mathar, Jul 24 2012
Also equals Gamma(1/3)*Gamma(2/3) = A073005 * A073006. - Jean-François Alcover, Nov 24 2014
From Amiram Eldar, Aug 06 2020: (Start)
Equals Integral_{x=0..oo} log(1 + 1/x^3) dx.
Equals Integral_{x=-oo..oo} exp(x/3)/(exp(x) + 1) dx. (End)
Equals Integral_{x=0..2*Pi} 1/(2 + sin(x)) dx; since for a>1: Integral_{x=0..2*Pi} 1/(a + sin(x)) dx = 2*Pi/sqrt(a^2-1). - Bernard Schott, Feb 18 2023
Equals 4*A093766. - Omar E. Pol, Dec 30 2023
From Stefano Spezia, Jun 05 2025: (Start)
Equals Beta(1/3,2/3).
Equals Integral_{x=-oo..oo} 1/(x^2 + x + 1) dx.
Equals 2*Integral_{x=0..oo} log(1 + x^3)/x^3 dx.
Equals Integral_{x=0..oo} log(1 + 4/(x*(x + 2))) dx. (End)

A256165 Decimal expansion of log(Gamma(1/3)).

Original entry on oeis.org

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

Offset: 0

Iaroslav V. Blagouchine, Mar 17 2015

			0.985420646927767069187174036977961391735556496385885...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A073005 (Gamma(1/3)), A256127 (second Malmsten integral), A256128 (third Malmsten integral).

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256166 (k=4), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

evalf(log(GAMMA(1/3)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Log[Gamma[1/3]],10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

log(gamma(1/3)) \\ Michel Marcus, Mar 17 2015

A034171 Related to triple factorial numbers A007559(n+1).

Original entry on oeis.org

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Wolfdieter Lang

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Michael De Vlieger, Table of n, a(n) for n = 0..1050
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
a(n) = A035529(n+1, 1) (first column of triangle).
Convolution of A004987(n) with A025748(n+1), n >= 0.
From R. J. Mathar, Jan 28 2020: (Start)
D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.
G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)
Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022
a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025

A202623 Decimal expansion of (1/3)! = Gamma(4/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.89297951156924921121856431365822588137622979265243370031680...

Index to sequences related to the Gamma function

Cf. A068467, A073005.

4^(8/9)*%PI^(2/3)*THETA[3](0,%E^-(16*%PI/SQRT(3)))^(2/3)/(3^(1/4)*(2^(7/16)*(SQRT(2)-1)^(1/4)/((SQRT(3)-1)^(1/8)*(SQRT(3)-SQRT(2))^(1/4))+1/(2^(1/4)*SQRT(SQRT(3)+1))+1)^(2/3))
/* This is exact, but degrades to 50+ digits if you replace
THETA[3](0,%E^-(16*%PI/SQRT(3)))
by 1+2*%E^-(16*%PI/SQRT(3)) */
/* R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011 */

Maple
```
evalf(GAMMA(4/3)) ;
```

RealDigits[(1/3)!,10,150][[1]] (* or *) RealDigits[Gamma[4/3],10,150] [[1]] (* Harvey P. Dale, Sep 03 2016 *)

A formula from R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011:
Equals (1/3) * (2*2^(7/9)*((Pi*EllipticTheta[3, 0, E^(-((16*Pi)/Sqrt[3]))])/ (1 + 1/(2^(1/4)*Sqrt[1 + Sqrt[3]]) + (2^(7/16)*((-1 + Sqrt[2])/(-Sqrt[2] + Sqrt[3]))^(1/4))/(-1+Sqrt[3])^(1/8)))^(2/3))/3^(1/4).
Equals Integral_{0..oo} exp(-x^3) dx. [Jean-François Alcover, Mar 29 2013]
Equals A073005/3. - R. J. Mathar, Jan 15 2021
Equals 3*Integral_{-1/e..0} (-LambertW(-1,x))^(1/3)-(-LambertW(x))^(1/3) dx. - Gleb Koloskov, Jun 07 2021

Corrected and extended by Harvey P. Dale, Sep 03 2016

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