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A328227
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Decimal expansion of positive solution to x^2 = 1 + (Pi + arccos(1/x))^2.
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1
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4, 6, 0, 3, 3, 3, 8, 8, 4, 8, 7, 5, 1, 7, 0, 0, 3, 5, 2, 5, 5, 6, 5, 8, 2, 0, 2, 9, 1, 0, 3, 0, 1, 6, 5, 1, 3, 0, 6, 7, 3, 9, 7, 1, 3, 4, 1, 6, 0, 5, 3, 2, 3, 4, 6, 0, 3, 9, 4, 3, 0, 1, 1, 5, 4, 3, 8, 4, 5, 8, 7, 3, 1, 9, 6, 5, 9, 7, 0, 9, 9, 8, 7, 1, 6, 5, 4, 6, 9, 9, 7, 2, 2, 7, 2
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OFFSET
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1,1
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COMMENTS
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We are in a rowboat on a circular lake, starting at the center. At the edge of the lake is a mean goblin. He can run k times as fast as we can row. This is the minimum value of k such that we will not be able to escape.
For a spirograph defined by complex function z = p * e^(-i * b * t) + b * e^(i * t), this is the value of p as b->oo such that each petal is tangent to the next one.
If we consider the set of all right triangles such that their tangent value is equal to the opposite angle in radians, this value is equal to the negative secant of the right triangle from that set with the smallest nonzero opposite angle. (End)
The envelope of the t*x = sin(t*y) family of curves contains the set of y = (-1)^n*k_n*x straight lines (n > 0), where k_n is the solution of (n*Pi + arccos(1/k))^2 + 1 = k^2. This entry is k_1. See illustration, section Links. - Luc Rousseau, Mar 11 2022
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LINKS
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FORMULA
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x=-sec(y), where decimal expansion of y is A115365.
Alternatively, x=sqrt(y^2+1).
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EXAMPLE
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4.6033388487517003525565820291030165130673971341605323460394301154384587319659...
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MATHEMATICA
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NSolve[x^2==1+(Pi+ArcCos[1/x])^2, x, Reals, WorkingPrecision->100]
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PROG
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(PARI) solve(x=4, 5, 1 + (Pi+acos(1/x))^2 - x^2) \\ Michel Marcus, Oct 08 2019
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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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