A073005 -id:A073005 - cp's OEIS frontend

A059551 Beatty sequence for Gamma(1/3).

2, 5, 8, 10, 13, 16, 18, 21, 24, 26, 29, 32, 34, 37, 40, 42, 45, 48, 50, 53, 56, 58, 61, 64, 66, 69, 72, 75, 77, 80, 83, 85, 88, 91, 93, 96, 99, 101, 104, 107, 109, 112, 115, 117, 120, 123, 125, 128, 131, 133, 136, 139, 141, 144, 147, 150, 152, 155, 158, 160, 163

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Beatty complement is A059552.

Cf. A073005 (Gamma(1/3)).

[Floor(n*Gamma(1/3)): n in [1..80]]; // Vincenzo Librandi, Jan 07 2015

Table[Floor[n Gamma[1/3]], {n, 70}] (* Vincenzo Librandi, Jan 07 2015 *)

{ default(realprecision, 100); b=gamma(1/3); for (n = 1, 2000, write("b059551.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*Gamma(1/3)). - Michel Marcus, Jan 04 2015

A059552 Beatty sequence for Gamma(1/3)/(Gamma(1/3)-1).

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 11, 12, 14, 15, 17, 19, 20, 22, 23, 25, 27, 28, 30, 31, 33, 35, 36, 38, 39, 41, 43, 44, 46, 47, 49, 51, 52, 54, 55, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73, 74, 76, 78, 79, 81, 82, 84, 86, 87, 89, 90, 92, 94, 95, 97, 98, 100, 102, 103, 105, 106, 108

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Beatty complement is A059551.

Cf. A073005.

[Floor(n*Gamma(1/3)/(Gamma(1/3)-1)): n in [1..80]]; // Vincenzo Librandi, Jan 06 2015

Floor[Range[100]*(1 + 1/(Gamma[1/3] - 1))] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=gamma(1/3)/(gamma(1/3) - 1); for (n = 1, 2000, write("b059552.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*Gamma(1/3)/(Gamma(1/3)-1)). - Michel Marcus, Jan 04 2015

A249206 Decimal expansion of the logarithmic capacity of the unit equilateral triangle.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 23 2014

			0.421753934648426824238122958592773059107710633283...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 4.9 Integer Chebyshev constants, p. 268.

Steven Finch, Electrical capacitance [Cached copy, with permission of the author]

Cf. A073005, A249205, A249220.

k = (Sqrt[3]/(8*Pi^2))*Gamma[1/3]^3; RealDigits[k, 10, 102] // First

k = (sqrt(3)/(8*Pi^2))*Gamma(1/3)^3.

A271923 Numerator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

Original entry on oeis.org

1, 5, 29, 52, 913, 1693, 69769, 658529, 1667651, 57873, 1616141, 1035959, 79918969, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

N. J. A. Sloane, May 04 2016

			1, 5/3, 29/13, 52/19, 913/285, 1693/465, 69769/17205, 658529/147963, 1667651/ 345247, 57873/11137, 1616141/291153, 1035959/175741, 79918969/12829093, ...

Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271924 (denominators), A073005, A186706.

f3:=proc(n) local j;
(1/3)*(mul(((2*j+1)*(3*j+4))/((j+1)*(6*j+1)),j=0..n-1)-1); end;
t3:=[seq(f3(n),n=1..50)];
map(numer,t3);
map(denom,t3);

a[n_] := (1/3)*(Product[((2*j + 1)*(3*j + 4))/((j + 1)*(6*j + 1)), {j, 0, n - 1}] - 1) // Numerator;
Array[a, 26] (* Jean-François Alcover, Nov 30 2017 *)

a(n)/A271924(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*sqrt(3)/(2*Pi) = A073005/A186706. - Amiram Eldar, Aug 17 2025

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).

Original entry on oeis.org

3, 5, 87, 156, 913, 1693, 69769, 658529, 5002953, 173619, 1616141, 3107877, 239756907, 3244922897, 3402714857, 6606018008, 51386679347, 5504537914811, 622652618545649, 10572475711004, 10931562934889, 235301799307039, 4608689892802861, 9034390134407023, 488936376609325, 959905250448181

Offset: 1

N. J. A. Sloane, May 04 2016

			3, 5, 87/13, 156/19, 913/95, 1693/155, 69769/5735, 658529/49321, 5002953/345247, 173619/11137, 1616141/97051, 3107877/175741, 239756907/12829093, ...

Jan de Gier, Loops, matchings and alternating-sign matrices, arXiv:math/0211285 [math.CO], 2002-2003.

Sequences of fractions from de Gier paper: A271919-A271926.

Cf. A271926 (denominators), A073005, A186706.

f3:=proc(n) local j;
(mul(((2*j+1)*(3*j+4))/((j+1)*(6*j+1)),j=0..n-1)-1); end;
t3:=[seq(f3(n),n=1..50)];
map(numer,t3);
map(denom,t3);

Table[Product[(2*j+1)*(3*j+4)/((j+1)*(6*j+1)),{j,0,n-1}]-1, {n,1,20}]//Numerator (* Vaclav Kotesovec, Oct 13 2017 *)

a(n)/A271926(n) ~ c * (2*n)^(2/3), where c = Gamma(1/3)*3^(3/2)/(2*Pi) = 3*A073005/A186706. - Amiram Eldar, Aug 17 2025

A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.61492662744600073515092236909361355359472818864859650504087875301429651...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 56, page 559.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.5.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), this sequence (Bi(0)), A358561 (Bi'(0)), A358564(Gi(0)).

RealDigits[AiryBi[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
airy(0)[2]
```
PARI
```
airy(0)[1]*sqrt(3)
```
PARI
```
3^(1/3)*gamma(1/3)/(2*Pi)
```

airy_bi(0).n(algorithm='scipy', prec=250)

Bi(0) = A284867*A002194.
Bi(0) = A358564*3.
Bi(0) = 1/(3^(1/6)*A073006).
Bi(0) = A073005/(3^(1/6)*A186706).
Bi(0) = A073005/(3^(1/6)*2*A093602).
Bi(0) = 3^(1/3)*A073005/(2*A000796).
Bi(0) = A252799/(3^(1/6)*BarnesG[5/3]).
Bi(0) = 3^(1/4)/(2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.44828835735382635791482371039882839086622679921226206108280877837233075...

F. W. J. Olver, Asymptotics and Special Functions, Academic Press, ISBN 978-0-12-525856-2, 1974.

[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.2.6.
Eric Weisstein's World of Mathematics, Airy Functions.
Wikipedia, Airy Function.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), this sequence (Bi'(0)), A358564 (Gi(0)).

RealDigits[AiryBi'[0], 10, 120][[1]] (* Amiram Eldar, Nov 28 2022 *)

PARI
```
derivnum(x=0, airy(x)[2])
```

airy_bi_prime(0).n(algorithm='scipy', prec=250)

Bi'(0) = A284868*A002194.
Bi'(0) = 3*Gi'(0), where Gi' is the derivative of the inhomogeneous Airy function of the first kind.
Bi'(0) = 3^(1/6)/A073005.
Bi'(0) = A073006*3^(1/6)/A186706.
Bi'(0) = A073006*3^(1/6)/2*A093602.
Bi'(0) = 3^(2/3)*A073006/(2*A000796).
Bi'(0) = 3^(1/4)*AGM(2,(sqrt(2+sqrt(3))))^(1/3)/(2^(7/9) * Pi^(2/3)), where AGM is the arithmetic-geometric mean.

A358564 Decimal expansion of Gi(0), where Gi is the inhomogeneous Airy function of the first kind (also called Scorer function).

Original entry on oeis.org

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

Offset: 0

Dumitru Damian, Nov 22 2022

			0.204975542482000245050307456364537851198242729549532168346959584338098839...

Scorer, R. S., Numerical evaluation of integrals of the form Integral_{x=x1..x2} f(x)*e^(i*phi(x))dx and the tabulation of the function Gi(z)=(1/Pi)*Integral_{u=0..oo} sin(u*z+u^3/3) du, Quart. J. Mech. Appl. Math. 3 (1950), 107-112.

Amparo Gil, Javier Segura, and Nico Temme, On nonoscillating integrals for computing inhomogeneous Airy functions, Mathematics of Computation 70.235 (2001): 1183-1194.
[DLMF] NIST Digital Library of Mathematical Functions, Eq. 9.12.6.
Allan J. MacLeod, Computation of inhomogeneous Airy functions, Journal of Computational and Applied Mathematics, Volume 53, Issue 1, 1994, Pages 109-116, ISSN 0377-0427.
Wikipedia, Scorer's function.
Index entries for transcendental numbers.

Cf. A284867 (Ai(0)), A284868 (Ai'(0)), A358559 (Bi(0)), A358561 (Bi'(0)), this sequence (Gi(0)).

First[RealDigits[N[ScorerGi[0],90]]] (* Stefano Spezia, Nov 28 2022 *)

PARI
```
airy(0)[2]/3
```
PARI
```
1/(3^(7/6)*gamma(2/3))
```
PARI
```
sqrt(3)*gamma(1/3)/(3^(7/6)*2*Pi)
```

1/(3^(3/4)*2^(2/9)*Pi^(1/3)*sqrtn(agm(2,(sqrt(2+sqrt(3)))),3))

1/(3^(7/6)*gamma(2/3)).n(algorithm='scipy', prec=250)

Gi(0) = A358559/3.
Gi(0) = A284867/A002194.
Gi(0) = Hi(0)/2, where Hi is the inhomogeneous Airy function of the second kind.
Gi(0) = 1/(3^(7/6)*A073006).
Gi(0) = A073005/(3^(7/6)*A186706).
Gi(0) = A073005/(3^(7/6)*2*A093602).
Gi(0) = A073005/(3^(4/6)*2*A000796).
Gi(0) = A252799/(3^(7/6)*BarnesG(5/3)).
Gi(0) = 1/(3^(3/4) * 2^(2/9) * Pi^(1/3) * AGM(2,(sqrt(2+sqrt(3))))^(1/3)), where AGM is the arithmetic-geometric mean.

A371881 Decimal expansion of Gamma(1/20).

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Apr 15 2024

			19.4700853112555128640473209677271275456304195833419756810827837553645...

Index to sequences related to gamma function

Cf. A073005, A175380, A256191, A203140, A371983.

evalf(GAMMA(1/20), 130);  # Alois P. Heinz, Apr 15 2024

RealDigits[Gamma[1/20], 10, 120][[1]]
RealDigits[2^(33/40) * 5^(5/16) * (1 + Sqrt[5])^(1/8) * Sqrt[5^(1/4) + Sqrt[2 + Sqrt[5]]] * Sqrt[Pi * Gamma[1/10]] * QPochhammer[E^(-2*Sqrt[5]*Pi)] / E^(Sqrt[5]*Pi/12), 10, 120][[1]]

Equals 2^(33/40) * 5^(5/16) * (1 + sqrt(5))^(1/8) * sqrt(5^(1/4) + sqrt(2 + sqrt(5))) * sqrt(Pi*Gamma(1/10)) * QPochhammer(exp(-2*sqrt(5)*Pi)) / exp(sqrt(5)*Pi/12).

A091546 First column of the array A092077 ((8,2)-Stirling2).

Original entry on oeis.org

1, 56, 10192, 3872960, 2517424000, 2497284608000, 3511182158848000, 6643156644540416000, 16275733779124019200000, 50129260039701979136000000, 189588861470152885092352000000, 863766852858016544480755712000000, 4666068539139005373285042356224000000, 29489553167358513959161467691335680000000

Offset: 1

Wolfdieter Lang, Feb 13 2004

Also seventh column (m=6) of triangle A091543.

Pawel Blasiak, Karol A. Penson, and Allan I. Solomon, The general boson normal ordering problem, Physics Letters A, Vol. 309, No. 3-4 (2003), pp. 198-205; arXiv preprint, arXiv:quant-ph/0402027, 2004.

Cf. A007559, A008542, A034689, A091543, A092077.

Cf. A073005, A175379.

a[n_] := 6^(2*n) * Pochhammer[1/6, n] * Pochhammer[1/3, n] / 2; Array[a, 20] (* Amiram Eldar, Aug 30 2025 *)

a(n) = (2^(n-1))*Product_{j=0..n-1} ((3*j+1)*(6*j+1)), n>=1. From eq.12 of the Blasiak et al. reference with r=8, s=2, k=1.
a(n) = (6^(2*n))*risefac(1/6, n)*risefac(1/3, n)/2, n>=1, with risefac(x, n) = Pochhammer(x, n).
a(n) = fac6(6*n-5)*fac6(6*n-4)/2, n>=1, with fac6(6*n-5) = A008542(n) and fac6(6*n-4)/2 = A034689(n)= (2^(n-1))*A007559(n), (6-factorials).
a(n) ~ Pi * (6/e)^(2*n) * n^(2*n-1/2) / (Gamma(1/6) * Gamma(1/3)). - Amiram Eldar, Aug 30 2025

cp's OEIS Frontend

A059551 Beatty sequence for Gamma(1/3).

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Magma

Mathematica

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A059552 Beatty sequence for Gamma(1/3)/(Gamma(1/3)-1).

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Magma

Mathematica

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A249206 Decimal expansion of the logarithmic capacity of the unit equilateral triangle.

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Mathematica

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A271923 Numerator of (1/3)*(Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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Maple

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A271925 Numerator of (Product_{j=0..n-1} (((2*j+1)*(3*j+4))/((j+1)*(6*j+1))) - 1).

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A358559 Decimal expansion of Bi(0), where Bi is the Airy function of the second kind.

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Mathematica

PARI

PARI

PARI

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Formula

A358561 Decimal expansion of the derivative Bi'(0), where Bi is the Airy function of the second kind.

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Examples

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Crossrefs

Programs

A271923 Numerator of (1/3)(Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6j+1))) - 1).

A271925 Numerator of (Product_{j=0..n-1} (((2j+1)(3j+4))/((j+1)(6*j+1))) - 1).