%I #34 Jan 01 2025 09:50:41

%S 1,2,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%T 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,

%U 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0

%N Decimal expansion of 1/8.

%C Also, decimal expansion of Integral_{x=0..oo} x^2/cosh(Pi*x) dx. - _Bruno Berselli_, Mar 20 2013

%C Also, decimal expansion of Sum_{i>=1} 1/9^i. - _Bruno Berselli_, Jan 03 2014

%C For any triangle ABC, sin(A/2) * sin(B/2) * sin(C/2) <= 1/8, equality is obtained only when the triangle is equilateral (see the Kiran S. Kedlaya link). - _Bernard Schott_, Sep 16 2022

%H Kiran S. Kedlaya, <a href="https://igor-kortchemski.perso.math.cnrs.fr/olympiades/Cours/ineqs-080299.pdf">A < B</a>, (1999), Problem 6.3, p. 6.

%H Srinivasa Ramanujan, <a href="https://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/question/q629.htm">Question 629</a>, Journal of the Indian Mathematical Society, Vol. 7 (1918), p. 40.

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (1).

%F Equals Sum_{k>=1} exp(-Pi*k^2) * (Pi*k^2 - 1/4) (Ramanujan, 1918). - _Amiram Eldar_, Jan 01 2025

%e 0.12500000000000000000...

%t PadRight[{1, 2, 5}, 100] (* _Paolo Xausa_, Aug 27 2024 *)

%Y Cf. A002194, A104956 (other trigonometric inequalities).

%K nonn,cons

%O 0,2

%A _N. J. A. Sloane_