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A248415
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Decimal expansion of the upper bound on length associated with the bow-and-arrow configuration used in the 2-arc smallest length problem.
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2
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4, 8, 1, 8, 9, 2, 6, 4, 5, 6, 3, 4, 5, 5, 7, 2, 8, 9, 3, 3, 2, 2, 4, 2, 6, 0, 9, 4, 8, 9, 1, 8, 8, 7, 1, 9, 1, 5, 1, 4, 5, 1, 5, 4, 6, 1, 8, 3, 4, 5, 3, 3, 4, 2, 6, 3, 6, 4, 3, 6, 2, 1, 9, 6, 9, 2, 5, 8, 2, 4, 0, 3, 4, 5, 0, 8, 7, 6, 6, 5, 8, 4, 6, 1, 0, 5, 8, 4, 1, 1, 8, 4, 4, 7, 3, 9, 2, 2, 1, 2, 1, 5, 7
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OFFSET
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1,1
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REFERENCES
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Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 8.11 Beam detection constants, p. 516.
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LINKS
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FORMULA
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L = 2*Pi - 2 theta_1 - theta_2 + 2*tan(theta_1/2) + sec(theta_2/2) - cos(theta_2/2) + tan(theta_1/2)*sin(theta_2/2), where theta_1 is A248413 and theta_2 A248414.
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EXAMPLE
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4.8189264563455728933224260948918871915145154618345334...
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MATHEMATICA
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x1 = Root[x^10 + 65*x^9 - 2139*x^8 + 20476*x^7 - 78054*x^6 + 126214*x^5 - 78054*x^4 + 20476*x^3 - 2139*x^2 + 65*x + 1, x, 4]; theta1 = 4*ArcTan[Sqrt[x1]]; theta2 = 2*ArcSin[2*Cos[theta1]]; L = 2*Pi - 2*theta1 - theta2 + 2*Tan[theta1/2] + Sec[theta2/2] - Cos[theta2/2] + Tan[theta1/2]*Sin[theta2/2]; RealDigits[L, 10, 103] // First
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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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