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%I #133 Nov 15 2023 12:26:32
%S 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,
%T 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,
%U 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
%N Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.
%C Only first term differs from the decimal expansion of phi.
%C 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)
%C 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
%C 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
%C An algebraic integer of degree 2, with minimal polynomial x^2 - 3x + 1. - _Charles R Greathouse IV_, Nov 12 2014 [The other root is 2 - phi = A132338 - _Wolfdieter Lang_, Aug 29 2022]
%C To eight digits: 5*(((Pi+1)/e)-1) = 2.61803395481182... - _Dan Graham_, Nov 21 2017
%C The ratio diagonal/side of the second smallest diagonal in a regular 10-gon. - _Mohammed Yaseen_, Nov 04 2020
%C 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
%D Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathématiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285.
%H Ivan Panchenko, <a href="/A104457/b104457.txt">Table of n, a(n) for n = 1..1000</a>
%H Murray Berg, <a href="http://www.fq.math.ca/Scanned/4-2/berg.pdf">Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers</a>, Fibonacci Quarterly, Vol. 4, No. 2 (1966), pp. 157-162.
%H John Hawkes et al., <a href="http://books.google.com/books?id=VPQ3AAAAMAAJ&pg=PA339&lpg=PA339#v=onepage&q&f=false">Question 1029</a>, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
%H Casey Mongoven, <a href="http://caseymongoven.com/catalogue/b12.html">Phi^2 number 1</a>; electronic music created using Phi^2.
%H Hideyuki Ohtsuka, <a href="https://www.fq.math.ca/Problems/ElemProbSolnNov2018.pdf">Problem B-1237</a>, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 4 (2018), p. 366; <a href="https://www.fq.math.ca/Problems/ElemProbSolnNov2019.pdf">A Telescoping Product</a>, Solution to Problem B-1237 by Steve Edwards, ibid., Vol. 57, No. 4 (2019), pp. 369-370.
%H Damien Roy, <a href="http://arxiv.org/abs/math/0303150">Diophantine Approximation in Small Degree</a>, arXiv:math/0303150 [math.NT], 2003.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciHyperbolicFunctions.html">Fibonacci Hyperbolic Functions</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticPolynomial.html">Chromatic Polynomial</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Perron_number">Perron number</a>.
%H Dmitrij Zelo, <a href="http://arxiv.org/abs/0903.0086">Simultaneous Approximation to Real and p-adic Numbers</a>, arXiv:0903.0086 [math.NT], 2009.
%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.
%H <a href="/index/Mo#moment_of_inertia">Index entries for sequences related to moment of inertia</a>.
%F Equals 2 + A094214 = 1 + A001622. - _R. J. Mathar_, May 19 2008
%F 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
%F Equals the nested radical sqrt(phi^2+sqrt(phi^4+sqrt(phi^8+...))). For a proof, see A094885. - _Stanislav Sykora_, May 24 2016
%F From _Christian Katzmann_, Mar 19 2018: (Start)
%F Equals Sum_{n>=0} (5*(2*n)!+8*n!^2)/(2*n!^2*3^(2*n+1)).
%F Equals 3/2 + Sum_{n>=0} 5*(2*n)!/(2*n!^2*3^(2*n+1)). (End)
%F Equals 1/A132338 = 2*A239798 = 5*A229780. - _Mohammed Yaseen_, Nov 04 2020
%F Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - _Amiram Eldar_, Dec 02 2021
%F c^n = phi * A001906(n) + A001519(n), where c = phi^2. - _Gary W. Adamson_, Sep 08 2023
%F Equals lim_{n->oo} S(n, 3)/S(n-1, 3) with the S-Chebyshev polynomials (see A049310), S(3, n) = A000045(2*(n+1)) = A001906(n+1). - _Wolfdieter Lang_, Nov 15 2023
%e 2.6180339887498948482045868343656381177203091798...
%t RealDigits[N[GoldenRatio+1,200]][[1]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 20 2011 *)
%o (PARI) (3+sqrt(5))/2 \\ _Charles R Greathouse IV_, Aug 21 2012
%o (Magma) SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // _G. C. Greubel_, Nov 23 2018
%o (Sage) numerical_approx(golden_ratio^2, digits=100) # _G. C. Greubel_, Nov 23 2018
%Y Cf. A001622, A094214, A094885, A132338, A229780, A239798, A341906.
%Y Cf. A001519, A001906, A049310.
%K nonn,cons,easy
%O 1,1
%A _Eric W. Weisstein_, Mar 08 2005
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