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A104457
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Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.
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57
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2, 6, 1, 8, 0, 3, 3, 9, 8, 8, 7, 4, 9, 8, 9, 4, 8, 4, 8, 2, 0, 4, 5, 8, 6, 8, 3, 4, 3, 6, 5, 6, 3, 8, 1, 1, 7, 7, 2, 0, 3, 0, 9, 1, 7, 9, 8, 0, 5, 7, 6, 2, 8, 6, 2, 1, 3, 5, 4, 4, 8, 6, 2, 2, 7, 0, 5, 2, 6, 0, 4, 6, 2, 8, 1, 8, 9, 0, 2, 4, 4, 9, 7, 0, 7, 2, 0, 7, 2, 0, 4, 1, 8, 9, 3, 9, 1, 1, 3, 7, 4, 8
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OFFSET
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1,1
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COMMENTS
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Only first term differs from the decimal expansion of phi.
Zelo extends work of D. Roy by showing that the square of the golden ratio is the optimal exponent of approximation by algebraic numbers of degree 4 with bounded denominator and trace. - Jonathan Vos Post, Mar 02 2009 (Cf. last sentence in the Zelo reference. - Joerg Arndt, Jan 04 2014)
Hawkes asks: "What two numbers are those whose product, difference of their squares, and the ratio or quotient of their cubes, are all equal to each other?". - Charles R Greathouse IV, Dec 11 2012
This is the case n=10 in (Gamma(1/n)/Gamma(3/n))*(Gamma((n-1)/n)/Gamma((n-3)/n)) = 1+2*cos(2*Pi/n). - Bruno Berselli, Dec 14 2012
To eight digits: 5*(((Pi+1)/e)-1) = 2.61803395481182... - Dan Graham, Nov 21 2017
The ratio diagonal/side of the second smallest diagonal in a regular 10-gon. - Mohammed Yaseen, Nov 04 2020
phi^2/10 is the moment of inertia of a solid regular icosahedron with a unit mass and a unit edge length (see A341906). - Amiram Eldar, Jun 08 2021
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REFERENCES
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Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 45.
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LINKS
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John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
Hideyuki Ohtsuka, Problem B-1237, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; A Telescoping Product, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
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FORMULA
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Satisfies these three equations: x-sqrt(x)-1 = 0; x-1/sqrt(x)-2 = 0; x^2-3*x+1 = 0. - Richard R. Forberg, Oct 11 2014
Equals the nested radical sqrt(phi^2+sqrt(phi^4+sqrt(phi^8+...))). For a proof, see A094885. - Stanislav Sykora, May 24 2016
Equals Sum_{n>=0} (5*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+1)).
Equals 3/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
Constant c = 2 + 2*cos(2*Pi/5).
The linear fractional transformation z -> c - c/z has order 5, that is, z = c - c/(c - c/(c - c/(c - c/(c - c/z)))). (End)
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EXAMPLE
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2.6180339887498948482045868343656381177203091798...
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MATHEMATICA
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PROG
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(Magma) SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // G. C. Greubel, Nov 23 2018
(Sage) numerical_approx(golden_ratio^2, digits=100) # G. C. Greubel, Nov 23 2018
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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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