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A069855
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Decimal expansion of the root of x*tan(x)=1.
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4
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8, 6, 0, 3, 3, 3, 5, 8, 9, 0, 1, 9, 3, 7, 9, 7, 6, 2, 4, 8, 3, 8, 9, 3, 4, 2, 4, 1, 3, 7, 6, 6, 2, 3, 3, 3, 4, 1, 1, 8, 8, 4, 3, 6, 3, 2, 3, 7, 6, 5, 3, 7, 8, 3, 0, 0, 3, 3, 8, 1, 2, 8, 5, 9, 0, 0, 4, 0, 3, 5, 5, 0, 7, 7, 2, 5, 8, 0, 2, 2, 1, 2, 3, 3, 4, 3, 0, 0, 8, 5, 7, 2, 1, 7, 1, 4, 2, 0, 8, 9, 1, 7, 4, 5
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OFFSET
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0,1
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COMMENTS
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Consider a lens-like shape S created by the curves cos(x) and -cos(x) for x in [-Pi/2,Pi/2] and the points A = (u, v), B = (-u, v), C = (-u, -v), D = (u, -v), K = (0, 2v), L = (-2u, 0), M = (0, -2v), N = (2u,0), where u is given by this sequence, and v = u/sqrt(1+u^2). Then ABCD is the rectangle of maximal area, inscribed in S, with sides parallel to the coordinate axes, and KLMN is the rhombus of minimal area, circumscribed around S, with vertices on the coordinate axes. Also, A,B,C,D are the tangent points where the sides of the rhombus touch S, see illustration in the links section. - Gleb Koloskov, Jul 05 2021
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LINKS
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FORMULA
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EXAMPLE
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0.860333589019379762483893424137662333411884363237653783...
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MATHEMATICA
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N[Minimize[{(x+Cot[x])^2 Sin[x], {x>0, x<Pi/2}}, x][[2]], 300][[1]][[2]] (* Gleb Koloskov, Jul 05 2021 *)
RealDigits[x/.FindRoot[x Tan[x]==1, {x, 1}, WorkingPrecision->120]][[1]] (* Harvey P. Dale, Dec 04 2021 *)
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PROG
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(PARI) /* 300 significant digits */ s=0.1; for(n=1, 500, s=s+sign(cotan(s)-s)/2^n; if(n>499, print(s*1.)))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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