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A346062
Decimal expansion of the minimum value of the area of a rhombus circumscribed around a cosine-shaped lens, whose vertices lie on coordinate axes.
2
4, 4, 8, 8, 7, 7, 0, 7, 0, 5, 5, 2, 8, 3, 6, 0, 5, 4, 0, 3, 2, 3, 2, 3, 0, 0, 2, 5, 2, 8, 9, 8, 1, 3, 6, 7, 0, 8, 8, 2, 2, 7, 9, 2, 4, 3, 6, 4, 4, 9, 2, 5, 7, 3, 6, 5, 4, 3, 6, 8, 3, 2, 3, 7, 4, 7, 9, 9, 0, 7, 8, 1, 8, 7, 4, 6, 6, 4, 5, 9, 3, 4, 0, 3, 7, 6, 1, 4, 9, 0, 7, 3, 5, 4, 4, 5, 5, 8, 3, 9, 4, 9, 9, 2
OFFSET
1,1
COMMENTS
Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and a rhombus circumscribed around S, whose vertices lie on coordinate axes.
This constant represents the value of the minimum area of such a rhombus KLMN with vertices K(0,2v), L(-2u,0), M(0,-2v), N(2u,0).
The rhombus touches S at the midpoints of its sides, A(u,v), B(-u,v), C(-u,-v), D(u,-v) which define a rectangle ABCD of the maximum area, inscribed in S, whose sides are parallel to coordinate axes. The constant u can be found as a root of equation x=cot(x) and is known as A069855, and v=cos(u)=u/sqrt(1+u^2).
FORMULA
Equals 8*A069855^2/sqrt(1+A069855^2).
EXAMPLE
4.4887707055283605403232300252898136708822792436449257365...
MATHEMATICA
N[Minimize[{2 (x+Cot[x])^2 Sin[x], {x>0, x<Pi/2}}, x], 120][[1]]
PROG
(PARI) u=solve(x=0.5, 1, x-cotan(x)); 8*u^2/sqrt(1+u^2)
CROSSREFS
Cf. A069855.
Sequence in context: A019674 A264606 A016712 * A333167 A246066 A076359
KEYWORD
nonn,cons
AUTHOR
Gleb Koloskov, Jul 03 2021
STATUS
approved