This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333322 #63 Mar 16 2025 10:43:52 %S A333322 6,4,9,5,1,9,0,5,2,8,3,8,3,2,8,9,8,5,0,7,2,7,9,2,3,7,8,0,6,4,7,0,2,1, %T A333322 3,7,6,0,3,5,5,1,9,7,0,1,7,8,8,9,2,7,3,5,5,2,0,9,2,7,6,1,7,2,9,4,4,7, %U A333322 4,8,8,1,3,4,0,8,0,0,0,1,3,9,0,5,4,2,9,8,2,0,0,3,3,9,6,8,2,1,5,8,7,8,3,5,9,8,0,3,0,3,0,7,7,7,5,1,3,6,3,6 %N A333322 Decimal expansion of (3/8) * sqrt(3). %C A333322 This is the area of the regular hexagon of diameter 1. %C A333322 From _Bernard Schott_, Apr 09 2022 and Oct 01 2022: (Start) %C A333322 For any triangle ABC, where (A,B,C) are the angles: %C A333322 sin(A) * sin(B) * sin(C) <= (3/8) * sqrt(3) [Bottema reference], %C A333322 cos(A/2) * cos(B/2) * cos(C/2) <= (3/8) * sqrt(3) [Mitrinovic reference], %C A333322 and if (ha,hb,hc) are the altitude lengths and (a,b,c) the side lengths of this triangle [Scott Brown link]: %C A333322 (ha+hb) * (hb+hc) * (hc+ha) / (a+b) * (b+c) * (c+a) <= (3/8) * sqrt(3). %C A333322 The equalities are obtained only when triangle ABC is equilateral. (End) %D A333322 O. Bottema et al., Geometric Inequalities, Groningen, 1969, item 2.7, page 19. %D A333322 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 207. %D A333322 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.15, p. 526. %D A333322 D. S. Mitrinovic, E. S. Barnes, D. C. B. Marsh, J. R. M. Radok, Elementary Inequalities, Tutorial Text 1 (1964), P. Noordhoff LTD, Groningen, problem 6.2.2, page 111. %H A333322 Scott Brown, <a href="https://cms.math.ca/wp-content/uploads/crux-pdfs/CRUXv36n5.pdf">Problem 3453</a>, Crux Mathematicorum, Vol. 36, No. 5 (2010), pp. 342 and 343. %F A333322 Equals A104954/2 or A104956/4. %e A333322 0.649519052838328985... %t A333322 RealDigits[(3/8) * Sqrt[3], 10, 120][[1]] %o A333322 (PARI) sqrt(27)/8 \\ _Charles R Greathouse IV_, Apr 09 2022 %Y A333322 Cf. A002194 (sqrt(3)), A104954. %Y A333322 Cf. A010527, A020821, A104956, A152623 (other geometric inequalities). %K A333322 cons,nonn %O A333322 0,1 %A A333322 _Kritsada Moomuang_, Mar 15 2020