A073005 -id:A073005 - cp's OEIS frontend

A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

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

Offset: 1

Jean-François Alcover, Jun 03 2014

			1.0174087975959560086695387533500634259952569...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.5.4 Gamma function, p. 34.

Eric Weisstein's MathWorld, Gamma function

Cf. A073005, A175379, A175574.

Re(evalf(4*EllipticK(sqrt((4*sqrt(3)-7)))/(sqrt(2+sqrt(3))*Pi), 120)); # Vaclav Kotesovec, Apr 22 2015

RealDigits[4*EllipticK[4*Sqrt[3]-7]/(Sqrt[2+Sqrt[3]]*Pi), 10, 103] // First
RealDigits[1/ArithmeticGeometricMean[1, Sqrt[2 + Sqrt[3]]/2], 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)
RealDigits[2 EllipticK[(2 - Sqrt[3])/4]/Pi, 10, 103][[1]] (* Jan Mangaldan, Jan 06 2017 *)

4*K(4*sqrt(3)-7)/(sqrt(2+sqrt(3))*Pi), where K is the complete elliptic integral of the first kind.
3^(1/4)*GAMMA(1/3)^3/(2*2^(1/3)*Pi^2), where GAMMA is the Euler Gamma function.
GAMMA(1/6)^(3/2)/(2^(5/6)*sqrt(3)*Pi^(5/4)).

A322508 Factorial expansion of Gamma(1/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

2, 1, 1, 0, 1, 2, 5, 6, 7, 2, 1, 8, 5, 7, 9, 12, 13, 10, 10, 13, 17, 18, 5, 1, 6, 3, 26, 13, 20, 29, 8, 31, 27, 19, 21, 27, 5, 14, 12, 3, 9, 37, 34, 40, 14, 29, 35, 12, 27, 4, 36, 22, 24, 11, 31, 37, 12, 5, 47, 14, 22, 18, 51, 20, 51, 4, 15, 54, 61, 26, 55, 2, 6, 73, 7, 17, 66, 54, 27

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			Gamma(1/3) = 2 + 1/2! + 1/3! + 0/4! + 1/5! + 2/6! + 5/7! + 6/8! + ...

Index entries for factorial base representation

Cf. A073005 (decimal expansion), A030651 (continued fraction).

Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322509 (Gamma(2/3)).

SetDefaultRealField(RealField(250));  [Floor(Gamma(1/3))] cat [Floor(Factorial(n)*Gamma(1/3)) - n*Floor(Factorial((n-1))*Gamma(1/3)) : n in [2..80]];

With[{b = Gamma[1/3]}, Table[If[n==1, Floor[b], Floor[n!*b] - n*Floor[(n-1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = gamma(1/3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=gamma(1/3);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Sep 23 2022

			0.9203713733179424976555185645431729947262...

Markus Faulhuber, An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a "Weltkonstante"-or-how Ramanujan split temperatures, The Ramanujan Journal volume 54, pages 1-27 (2021). See p. 23.
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 2.

Cf. A004117.
Cf. A081760.
Cf. A073005, A175379, A203145.

First[RealDigits[N[Gamma[1/6]/(2Gamma[1/3]Gamma[5/6]),88]]]

1/(2*gamma(1/3)*gamma(5/6)/gamma(1/6)) \\ Michel Marcus, Sep 24 2022

Equals 1/(2*A081760) = A175379/(2*A073005*A203145).
Equals Sum_{k,m in Z^2} exp(-Pi*(2/sqrt(3))*(k^2+k*m+m^2))*exp(2*Pi*i*(k/3-m/3)).
Equals Sum_{k>=0} (binomial(-1/3,2*k)^2 - binomial(-1/3,2*k+1)^2). - Gerry Martens, Jul 24 2023
Equals 3*Gamma(1/3)^3 / (2^(8/3) * Pi^2). - Vaclav Kotesovec, Jul 27 2023

A371983 Decimal expansion of Gamma(1/30).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			29.4547797456996940196962082886383457347018736055729711046565415567498...

Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 2009.
R. Vidunas, Expressions for values of the Gamma function, arxiv:math/0403510 [math.CA], 2004.
Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371881.

evalf(GAMMA(1/30), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/30], 10, 120][[1]]
RealDigits[2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma[1/5] * Gamma[1/3] / ((10 + Sqrt[5] - Sqrt[75 + 30*Sqrt[5]])^(1/4) * Sqrt[Pi]), 10, 120][[1]]

Equals 3^(9/20) * sqrt(5 + sqrt(5)) * sqrt(sqrt(15) + sqrt(5 + 2*sqrt(5))) * Gamma(1/3) * Gamma(1/5) / (sqrt(Pi) * 2^(16/15) * 5^(1/6)).
Equals 2^(11/60) * 3^(9/20) * 5^(1/3) * Gamma(1/5) * Gamma(1/3) / ((10 + sqrt(5) - sqrt(75 + 30*sqrt(5)))^(1/4) * sqrt(Pi)).
Equals 8*Pi^2 / (Gamma(17/30) * Gamma(19/30) * Gamma(23/30)).
Equals Gamma(7/30) * Gamma(11/30) * Gamma(13/30) / (2*Pi*A019815).

A377008 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(2/3)^(2*k).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Oct 12 2024

			0.88310581396712625588502373888562329082705924490169...

H. M. Srivastava and Junesang Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Insights, 2011, p. 272, eq. (29).

Eric Weisstein's MathWorld, Riemann Zeta Function.
Wikipedia, Riemann zeta function.

Cf. A073005, A203129, A256923, A275486.

RealDigits[Log[4*Pi/(3*Sqrt[3])], 10, 120][[1]]
(* or *)
RealDigits[Log[Gamma[1/3]*Gamma[5/3]], 10, 120][[1]]

PARI
```
log(4*Pi/(3*sqrt(3)))
```
PARI
```
log(gamma(1/3)*gamma(5/3))
```

Equals log(4*Pi/(3*sqrt(3))) = log(A275486).

Equals log(Gamma(1/3)*Gamma(5/3)).

A059188 Engel expansion of Gamma(1/3) = 2.6789385....

Original entry on oeis.org

1, 1, 2, 3, 14, 33, 57, 236, 6280, 7170, 172302, 24568434, 32871132, 43231756, 60680523, 83128444, 720494727, 803406064, 1804216488, 6655647717, 9106036988, 14962799365, 37839117297, 121819278396, 262108609568

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A006784, A073005.

EngelExp[ A_, n_ ] := Join[ Array[ 1&, Floor[ A ] ], First@Transpose@NestList[ {Ceiling[ 1/Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ] ], Expand[ #[ [ 1 ] ]#[ [ 2 ] ]-1 ]}&, {Ceiling[ 1/(A-Floor[ A ]) ], A-Floor[ A ]}, n-1 ] ]

A113273 Decimal expansion of Gamma(1/3)^3/Pi^2.

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Jan 07 2006

Known to be transcendental.

			1.9479979816084244775451641...

Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008. (hal-00407231)
Index entries for transcendental numbers.

Cf. A002388, A073005.

RealDigits[Gamma[1/3]^3/Pi^2, 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

PARI
```
gamma(1/3)^3/Pi^2
```

Offset corrected by Sean A. Irvine, May 24 2025

A242010 Decimal expansion of sum_{k>=0} (log(3k+1)/(3k+1)-log(3k+2)/(3k+2)).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 11 2014

			-0.22266298696860150948666026276474436188657161605715247851290026...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 8.

Cf. A073005, A073006.

RealDigits[(Pi/Sqrt[3])*(Log[Gamma[1/3]/Gamma[2/3]] - (1/3)*(EulerGamma + Log[2*Pi])), 10, 104] // First

(Pi/sqrt(3))*(log(Gamma(1/3)/Gamma(2/3)) - (1/3)*(gamma + log(2*Pi))), where gamma is Euler's constant and Gamma(x) is the Euler Gamma function.

A255902 Decimal expansion of the limit as n tends to infinity of n*s_n, where the s_n are the hexagonal circle-packing rigidity constants.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Mar 10 2015

			4.4516506980892215382479987827401255099693875...

P. Doyle, Zheng-Xu He, and B. Rodin, The asymptotic value of the circle-packing rigidity constants, Discrete Comput. Geom. 12 (1994).
Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 68.
Eric Weisstein's MathWorld, Conformal Radius
Wikipedia, Circle packing theorem

Cf. A073005 (gamma(1/3)), A073006 (gamma(2/3)).

RealDigits[(2^(4/3)/3)*Gamma[1/3]^2/Gamma[2/3], 10, 102] // First

(2^(4/3)/3)*gamma(1/3)^2/gamma(2/3).

Equals 4/R, where R = 2^(2/3)*gamma(2/3)/(gamma(1/3)*gamma(4/3)) is the conformal radius in a mapping from the unit disk to the unit side hexagon satisfying certain conditions.

A273842 Decimal expansion of the double integral int_{0..inf} int_{0..inf} 1/sqrt((1+x^2)(1+y^2)(1+(x+y)^2)) dx dy.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 01 2016

			2.94917198474238496070570479120917479184367657310611674089147554...

David H. Bailey, Jonathan M. Borwein, David Broadhurst and M. L. Glasser, Elliptic integral evaluations of Bessel moments, arXiv:0801.0891 [hep-th], 2008.

Cf. A073005.

RealDigits[ Gamma[1/3]^6/(8*2^(2/3)*Pi^2) , 10, 103][[1]]

gamma(1/3)^6/(8*2^(2/3)*Pi^2) \\ Michel Marcus, Jun 01 2016

Gamma(1/3)^6/(8*2^(2/3)*Pi^2).

cp's OEIS Frontend

A243308 Decimal expansion of h_3, a constant related to certain evaluations of the gamma function from elliptic integrals.

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A322508 Factorial expansion of Gamma(1/3) = Sum_{n>=1} a(n)/n!.

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A357318 Decimal expansion of 1/(2*L), where L is the conjectured Landau's constant A081760.

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A371983 Decimal expansion of Gamma(1/30).

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A377008 Decimal expansion of Sum_{k>=1} (zeta(2*k)/k)*(2/3)^(2*k).

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A059188 Engel expansion of Gamma(1/3) = 2.6789385....

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A113273 Decimal expansion of Gamma(1/3)^3/Pi^2.

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A377008 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)(2/3)^(2*k).