A229780 -id:A229780 - cp's OEIS frontend

A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

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

Offset: 1

Eric W. Weisstein, Mar 08 2005

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
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]
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

			2.6180339887498948482045868343656381177203091798...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.17.1, p. 153.
Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 138-139.
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.

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Murray Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, Vol. 4, No. 2 (1966), pp. 157-162.
John Hawkes et al., Question 1029, The Mathematical Questions Proposed in the Ladies' Diary (1817), p. 339. Originally published 1798 and answered in 1799.
Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.
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.
Damien Roy, Diophantine Approximation in Small Degree, arXiv:math/0303150 [math.NT], 2003.
Eric Weisstein's World of Mathematics, Chromatic Polynomial.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions.
Wikipedia, Perron number.
Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, arXiv:0903.0086 [math.NT], 2009.
Index entries for algebraic numbers, degree 2.
Index entries for sequences related to moment of inertia.

Cf. A001622, A094214, A094885, A132338, A229780, A239798, A341906.
Cf. A001519, A001906, A032908, A049310.
2 + 2*cos(2*Pi/n): A116425 (n = 7), A332438 (n = 9), A296184 (n = 10), A019973 (n = 12).

SetDefaultRealField(RealField(100)); (1+Sqrt(5))^2/4; // G. C. Greubel, Nov 23 2018

RealDigits[N[GoldenRatio+1,200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

(3+sqrt(5))/2 \\ Charles R Greathouse IV, Aug 21 2012

numerical_approx(golden_ratio^2, digits=100) # G. C. Greubel, Nov 23 2018

Equals 2 + A094214 = 1 + A001622. - R. J. Mathar, May 19 2008
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
From Christian Katzmann, Mar 19 2018: (Start)
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 1/A132338 = 2*A239798 = 5*A229780. - Mohammed Yaseen, Nov 04 2020
Equals Product_{k>=1} 1 + 1/(phi + phi^k), where phi is the golden ratio (A001622) (Ohtsuka, 2018). - Amiram Eldar, Dec 02 2021
c^n = phi * A001906(n) + A001519(n), where c = phi^2. - Gary W. Adamson, Sep 08 2023
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
From Peter Bala, May 08 2024: (Start)
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)
Equals Product_{k>=1} (1 + 1/A032908(k)). - Amiram Eldar, Nov 28 2024

A242671 Decimal expansion of k2, a Diophantine approximation constant such that the area of the "critical parallelogram" (in this case a square) is 4*k2.

Original entry on oeis.org

7, 2, 3, 6, 0, 6, 7, 9, 7, 7, 4, 9, 9, 7, 8, 9, 6, 9, 6, 4, 0, 9, 1, 7, 3, 6, 6, 8, 7, 3, 1, 2, 7, 6, 2, 3, 5, 4, 4, 0, 6, 1, 8, 3, 5, 9, 6, 1, 1, 5, 2, 5, 7, 2, 4, 2, 7, 0, 8, 9, 7, 2, 4, 5, 4, 1, 0, 5, 2, 0, 9, 2, 5, 6, 3, 7, 8, 0, 4, 8, 9, 9, 4, 1, 4, 4, 1, 4, 4, 0, 8, 3, 7, 8, 7, 8, 2, 2, 7, 4

Offset: 0

Jean-François Alcover, May 20 2014

Quoting Steven Finch: "The slopes of the 'critical parallelogram' are (1+sqrt(5))/2 [phi] and (1-sqrt(5))/2 [-1/phi]."
Essentially the same as A229780, A134972, A134945, A098317 and A002163. - R. J. Mathar, May 23 2014
Let W_n be the collection of all binary words of length n that do not contain two consecutive 0's. Let r_n be the ratio of the total number of 1's in W_n divided by the total number of letters in W_n. Then lim_{n->oo} r_n = 0.723606... Equivalently, lim_{n->oo} A004798(n)/(n*A000045(n+2)) = 0.723606... - Geoffrey Critzer, Feb 04 2022
The limiting frequency of the digit 0 in the base phi representation of real numbers in the range [0,1], where phi is the golden ratio (A001622) (Rényi, 1957). - Amiram Eldar, Mar 18 2025

			k2 = 0.723606797749978969640917366873127623544...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.23, p. 176.

Alfréd Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar., Vol.8, No. 3-4 (1957), pp. 477-493.
Index entries for algebraic numbers, degree 2.

Cf. A000032, A000045, A001622, A002163, A004798, A020762, A081007, A094214, A094874, A098317, A134972, A229780, A244847, A296184, A300074, A344212.

RealDigits[(1+1/Sqrt[5])/2, 10, 100] // First

(1 + 1/sqrt(5))/2 \\ Stefano Spezia, Dec 07 2024

Equals (1 + 1/sqrt(5))/2.
Equals 1/A094874. - Michel Marcus, Dec 01 2018
From Amiram Eldar, Feb 11 2022: (Start)
Equals phi/sqrt(5), where phi is the golden ratio (A001622).
Equals lim_{k->oo} Fibonacci(k+1)/Lucas(k). (End)
From Amiram Eldar, Nov 28 2024: (Start)
Equals A344212/2 = A296184/5 = A300074^2 = sqrt(A229780).
Equals Product_{k>=1} (1 - 1/A081007(k)). (End)
Equals 1 - A244847. - Amiram Eldar, Mar 18 2025

cp's OEIS Frontend

A104457 Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.

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