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A139339
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Decimal expansion of the square root of the golden ratio.
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30
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1, 2, 7, 2, 0, 1, 9, 6, 4, 9, 5, 1, 4, 0, 6, 8, 9, 6, 4, 2, 5, 2, 4, 2, 2, 4, 6, 1, 7, 3, 7, 4, 9, 1, 4, 9, 1, 7, 1, 5, 6, 0, 8, 0, 4, 1, 8, 4, 0, 0, 9, 6, 2, 4, 8, 6, 1, 6, 6, 4, 0, 3, 8, 2, 5, 3, 9, 2, 9, 7, 5, 7, 5, 5, 3, 6, 0, 6, 8, 0, 1, 1, 8, 3, 0, 3, 8, 4, 2, 1, 4, 9, 8, 8, 4, 6, 0, 2, 5, 8, 5, 3, 8, 5, 1
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OFFSET
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1,2
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COMMENTS
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The hyperbolas x^2 - y^2 = 1 and xy = 1 meet at (c, 1/c) and (-c, -1/c), where c = sqrt(golden ratio); see the Mathematica program for a graph. - Clark Kimberling, Oct 19 2011
Also the limiting value of the ratio of the slopes of the tangents drawn to the function y=sqrt(x) from the abscissa F(n) points (where F(n)=A000045(n) are the Fibonacci numbers and n > 0). - Burak Muslu, Apr 04 2021
The length of the base of the isosceles triangle of smallest perimeter which circumscribes a unit-diameter semicircle (DeTemple, 1992). - Amiram Eldar, Jan 22 2022
The unique real solution to arcsec(x) = arccot(x). - Wolfe Padawer, Apr 14 2023
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REFERENCES
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B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 45-48.
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LINKS
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Mohammad K. Azarian, Problem 123, Missouri Journal of Mathematical Sciences, Vol. 10, No. 3 (Fall 1998), p. 176. Solution published in Vol. 12, No. 1 (Winter 2000), pp. 61-62.
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FORMULA
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Equals sqrt((1 + sqrt(5))/2).
Equals tan(arccos(1/phi)).
Equals cot(arcsin(1/phi)). (End)
Equals 5^(1/4)*cos(arctan(2)/2).
Equals Re(sqrt(1+2*i)) (the imaginary part is A197762). (End)
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EXAMPLE
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1.2720196495140689642524224617374914917156080418400...
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MAPLE
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Digits:=100: evalf(sqrt((1+sqrt(5))/2)); # Muniru A Asiru, Sep 11 2018
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MATHEMATICA
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N[Sqrt[GoldenRatio], 100]
FindRoot[x*Sqrt[-1 + x^2] == 1, {x, 1.2, 1.3}, WorkingPrecision -> 110]
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PROG
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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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