A113477 - OEIS

%I #23 Apr 22 2024 04:58:27

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

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

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

%N Decimal expansion of Gamma(1/3)^3/(2^(4/3)*Pi).

%C This number is transcendental from a result of Schneider on elliptic integrals.

%H G. C. Greubel, <a href="/A113477/b113477.txt">Table of n, a(n) for n = 0..5000</a>

%H Th. Schneider, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002173042">Transzendenzuntersuchungen periodischer Funktionen</a>, Journal für die reine und angewandte Mathematik (1935) Volume: 172, page 65-74.

%H Th. Schneider, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002278537">Arithmetische Untersuchungen elliptischer Integrale</a>, Mathematische Annalen (1937) Volume: 113, page I-XIII.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%F Equals Integral_{x>=1} dx/sqrt(4*x^3-4).

%F Equals 2*Integral_{x=0..1} dx/sqrt(1-x^6). - _Takayuki Tatekawa_, Apr 15 2024

%F Equals Beta(1/6, 1/2) / 3. - _Peter Luschny_, Apr 15 2024

%e 2.428650647887581611819....

%p Beta(1/6, 1/2)/3: evalf(%, 106); # _Peter Luschny_, Apr 15 2024

%t RealDigits[Gamma[1/3]^3/(Pi*2^(4/3)), 10, 5001][[1]] (* _G. C. Greubel_, Mar 12 2017 *)

%o (PARI) gamma(1/3)^3/2^(4/3)/Pi

%Y Cf. A085565.

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Jan 08 2006