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A290570
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Decimal expansion of Integral_{0..Pi/2} dtheta/(cos(theta)^3 + sin(theta)^3)^(2/3).
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4
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1, 7, 6, 6, 6, 3, 8, 7, 5, 0, 2, 8, 5, 4, 4, 9, 9, 5, 7, 3, 1, 3, 6, 8, 9, 4, 9, 9, 6, 4, 8, 4, 3, 8, 7, 0, 2, 5, 7, 1, 8, 6, 8, 5, 3, 8, 2, 0, 2, 5, 5, 7, 5, 3, 0, 1, 2, 6, 9, 0, 5, 2, 4, 1, 8, 3, 5, 4, 5, 3, 0, 0, 1, 7, 2, 8, 1, 0, 7, 9, 1, 3, 6, 0, 5, 4, 8, 6, 9, 9, 3, 3, 3, 3, 3, 8, 3, 5, 8, 7, 2, 1, 9, 3, 4
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OFFSET
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1,2
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REFERENCES
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Oscar S. Adams, Elliptic Functions Applied to Conformal World Maps, Special Publication No. 112 of the U.S. Coast and Geodetic Survey, 1925. See constant K p. 9 and previous pages.
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LINKS
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StackExchange, Question 2380131 Evaluate in closed form Integral_{0..Pi/2} dtheta/(cos(theta)^3+sin(theta)^3)^(2/3), August 2017.
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FORMULA
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Equals (1/3)*Beta(1/3,1/3).
Equals (1/3)*Gamma(1/3)^2/Gamma(2/3).
Equals 2*Sum_{n >= 0} (1/(3*n+1) + 1/(3*n-2))*binomial(1/3,n). Cf. A002580 and A175576.
Equals Sum_{n >= 0} (-1)^n*(1/(3*n+1) - 1/(3*n-2))*binomial(1/3,n).
Equals hypergeom([1/3, 2/3], [4/3], 1) = (3/2)*hypergeom([-1/3, -2/3], [4/3], 1) = 2*hypergeom([1/3, 2/3], [4/3], -1) = hypergeom([-1/3, -2/3, 5/6], [4/3, -1/6], -1). (End)
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EXAMPLE
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1.766638750285449957313689499648438702571868538202557530126905241835453...
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MATHEMATICA
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RealDigits[(1/3)*Gamma[1/3]^2/Gamma[2/3], 10, 105]
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PROG
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(PARI) (1/3)*gamma(1/3)^2/gamma(2/3) \\ Michel Marcus, Aug 07 2017
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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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