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Decimal expansion of 5*sqrt(2*Pi)/4.
2

%I #24 Jun 13 2023 04:31:49

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

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

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

%N Decimal expansion of 5*sqrt(2*Pi)/4.

%C Upper bound using Shannon entropy arising in randomly-projected hypercubes.

%H David L. Donoho and Jared Tanner, <a href="https://doi.org/10.1007/s00454-009-9221-z">Counting the faces of randomly-projected hypercubes and orthants, with applications</a>, Discrete & computational geometry, Vol. 43 (2010), pp. 522-541, see p. 533; <a href="http://arxiv.org/abs/0807.3590">arXiv preprint</a>, arXiv:0807.3590 [math.MG], 2008, see p. 10.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fresnel_integral">Fresnel Integral</a>.

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

%F Equals 10*Integral_{x>=0} x*sin(x^4) dx or 10*Integral_{x>=0} x*cos(x^4) dx (Fresnel integrals).

%e 3.13328534328875...

%t RealDigits[5*Sqrt[2*Pi]/4, 10, 120][[1]] (* _Amiram Eldar_, Jun 13 2023 *)

%o (PARI) 5*sqrt(2*Pi)/4 \\ _Michel Marcus_, Mar 06 2020

%Y Apart from possible scaling sqrt(A019692/2^n) for n=0..7 are A019727, A002161, A069998, A019704, A217481, A019706, this sequence, A019710.

%Y Cf. A143148 (lower bound).

%K cons,easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Jul 27 2008

%E Edited and a(100) corrected by _Georg Fischer_, Jul 16 2021