A146753 -id:A146753 - cp's OEIS frontend

A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

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

Offset: 1

Eric W. Weisstein, Apr 22 2006

General formula: Integral_{x=0..1} (1+x^(3n))/sqrt(1-x^3) dx = G_3 * k_n = G_3*A146751(n)/A146752(n) = A118292*A146751(n)/A146752(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi) is the number in the present entry. For numerators of k_n see A146752, for denominators of k_n see A146753. - Artur Jasinski

gamma(1/6)*gamma(1/3)/(3*sqrt(Pi)) = gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi). - Harry J. Smith, May 09 2009

			2.8043642106509085223500381581005882709260444108....

Harry J. Smith, Table of n, a(n) for n = 1..4000
Eric Weisstein's World of Mathematics, Butterfly Curve

Cf. A146752, A146753, A160323 (continued fraction).

RealDigits[(Gamma[1/3]^3)/(2^(1/3) Sqrt[3] Pi), 10, 200] (* Artur Jasinski*)

allocatemem(932245000); default(realprecision, 4080); x=gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi); for (n=1, 4000, d=floor(x); x=(x-d)*10; write("b118292.txt", n, " ", d)) \\ Harry J. Smith, Jun 20 2009

3/hypergeom([1/3,1/6],[3/2],1) \\ Charles R Greathouse IV, Aug 29 2025

Equals A073005^3 / (A002194*A002580*A000796) [see Vidunas, arXiv:math.CA/0403510]. - R. J. Mathar, Nov 30 2008

Equals 3/hypergeom([1/3, 1/6], [3/2], 1) = A290570*A005480. - Peter Bala, Mar 02 2022

Edited by N. J. A. Sloane, Nov 16 2008 at the suggestion of R. J. Mathar

A146752 a(n) = numerator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

Original entry on oeis.org

1, 7, 71, 1159, 5197, 148025, 730141, 29616293, 125438657, 1319937329, 77390680651, 76972298827, 319946679037, 3504590799071, 289784158718029, 25703039917515461, 1114069690728835, 112203290640603311

Offset: 0

Artur Jasinski, Nov 01 2008

Previous name was: a(n) is the numerator of k_n such that Integral_{x=0..1} ((1+x^(3n))/sqrt(1-x^3)) dx = k_n*Gamma(1/3)^3/(2^(1/3)*sqrt(3)*Pi) for n >= 0.

General formula: Integral_{x=0..1} ((1+x^(3n))/sqrt(1-x^3)) dx = G_3 * k_n = G_3*A146752(n)/A146753(n) = A118292*A146752(n)/A146753(n) where G_3 = (Gamma(1/3)^3)/(2^(1/3)*sqrt(3)*Pi).

Cf. A146753 (denominator), A118292 (G_3).

Table[Numerator[(1/2) (1 + Product[(2 (1 + 3 k))/(5 + 6 k), {k, 0, n - 1}])], {n, 0, 30}]

a(n) = numerator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

Simpler name (using given formula) from Joerg Arndt, Sep 24 2022

cp's OEIS Frontend

A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

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A146752 a(n) = numerator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).

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cp's OEIS Frontend

A118292 Decimal expansion of (Gamma(1/6)*Gamma(1/3))/(3*sqrt(Pi)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A146752 a(n) = numerator((1/2)*(1 + Product_{k=0..n-1} 2*(1 + 3*k)/(5 + 6*k))).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

Extensions

A118292 Decimal expansion of (Gamma(1/6)Gamma(1/3))/(3sqrt(Pi)).

A146752 a(n) = numerator((1/2)(1 + Product_{k=0..n-1} 2(1 + 3k)/(5 + 6k))).