A073006 -id:A073006 - cp's OEIS frontend

A256923 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(1/3)^(2k).

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

Offset: 0

Jean-François Alcover, Apr 13 2015

			0.189958633407180946467791617427446722751559110541442648...

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

Eric Weisstein's MathWorld, Riemann Zeta Function
Wikipedia, Riemann Zeta Function

Cf. A073006, A202623, A248897, A256924.

RealDigits[Log[2*Pi/(3*Sqrt[3])], 10, 105] // First

Equals log(Gamma(2/3)*Gamma(4/3)).
Equals log(2*Pi/(3*sqrt(3))).
Equals log(A248897).
Equals -Sum_{k>=1} log(1 - 1/(3*k)^2). - Amiram Eldar, Aug 12 2020

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.

A203129 Decimal expansion of (2/3)! = Gamma(5/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 29 2011

			0.90274529295093361129685868543634252367955151070452913226268...

Index to sequences related to the Gamma function.

Cf. A073006.

RealDigits[Gamma[5/3], 10, 120][[1]] (* Amiram Eldar, May 29 2023 *)

gamma(5/3) \\ Charles R Greathouse IV, Mar 25 2014

Equals 2*A073006/3. - R. J. Mathar, Jan 15 2021
Equals Integral_{x=0..oo} exp(-x^(3/2)) dx. - Ilya Gutkovskiy, Apr 10 2024
Equals sqrt(Pi)*Gamma(7/3)/(2^(4/3)*Gamma(7/6)). - Gerry Martens, May 01 2024
Equals 4*Pi / (3^(3/2) * Gamma(1/3)). - Vaclav Kotesovec, May 02 2024

A218540 Reduced third-order Patalan numbers.

Original entry on oeis.org

1, 1, 1, 5, 10, 66, 154, 1122, 2805, 21505, 55913, 442221, 1179256, 9524760, 25852920, 211993944, 582983346, 4835332458, 13431479050, 112400272050, 314720761740, 2652646420380, 7475639911980, 63380425340700, 179577871798650, 1530003467724498

Offset: 0

R. J. Mathar, Nov 01 2012

Obtained by removing powers of 3 in a systematic manner from the Patalan numbers A025748.

Richard J. Mathar, Series expansion of generalized Fresnel integrals, arXiv:1211.3963 [math.CA], 2012, eq. (4.19).

Cf. A025748, A073006, A108411.

A218540 := proc(n)
    option remember;
    if n <=2 then
        1;
    elif n = 3 then
        5 ;
    else
        (n-1)*(n-2)*(n+4)*procname(n-1)-3*(3*n-4)*(3*n-7)*(n+2)*procname(n-2)-3*(3*n-10)*(n+4)*(3*n-7)*procname(n-3) ;
        -%/n/(n+2)/(n-1) ;
    end if;
end proc:

a[n_] := 3^(2*n-2-Floor[n/2]) * Pochhammer[2/3, n-1]/n!; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n) = A025748(n)/A108411(n).
D-finite with recurrence n*(n+2)*(n-1)*a(n) + (n-1)*(n-2)*(n+4)*a(n-1) - 3*(3*n-4)*(3*n-7)*(n+2)*a(n-2) - 3*(3*n-10)*(n+4)*(3*n-7)*a(n-3) = 0, n >= 4.
a(n) ~ 3^(2*n-2-floor(n/2)) / (Gamma(2/3) * n^(4/3)). - Amiram Eldar, Aug 20 2025

A322509 Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 6, 6, 0, 3, 1, 11, 7, 6, 6, 14, 1, 8, 12, 15, 8, 17, 8, 1, 13, 15, 3, 4, 10, 16, 25, 1, 25, 22, 6, 3, 19, 17, 8, 10, 25, 37, 29, 17, 35, 19, 24, 25, 30, 31, 4, 7, 51, 49, 14, 51, 45, 54, 0, 26, 34, 41, 56, 57, 16, 15, 63, 4, 51, 42, 13, 35, 12, 15, 66, 22, 13, 43, 14, 78

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

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

Index entries for factorial base representation

Cf. A073006 (decimal expansion), A030652 (continued fraction).

Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322508 (Gamma(1/3)).

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

With[{b = Gamma[2/3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = gamma(2/3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=gamma(2/3);
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)]

A059189 Engel expansion of Gamma(2/3) = 1.35412.

Original entry on oeis.org

1, 3, 17, 17, 50, 79, 796, 3687, 7074, 9098, 95915, 118514, 133188, 186305, 209314, 666015, 5735240, 7685979, 11174747, 97173279, 269061009, 569125952, 932655002, 7282946876, 9919537325, 52110120678, 70254144261

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.

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.
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.
Index entries for sequences related to Engel expansions

Cf. A073006.

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 ] ]

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.

cp's OEIS Frontend

A256923 Decimal expansion of Sum_{k>=1} (zeta(2k)/k)*(1/3)^(2k).

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

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

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

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Mathematica

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A203129 Decimal expansion of (2/3)! = Gamma(5/3).

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A218540 Reduced third-order Patalan numbers.

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A322509 Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

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