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A104457
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Decimal expansion of 1 + phi = 1 + Golden Ratio (cf. A001622).
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23
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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
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Equivalently, decimal expansion of phi^2.
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. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 02 2009]
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REFERENCES
| M. Berg, Phi, the golden ratio (to 4599 decimal places) and Fibonacci numbers, Fibonacci Quarterly, 4 (1961), 157-162.
Damien Roy. Diophantine Approximation in Small Degree. Centre de Recherches Mathematiques. CRM Proceedings and Lecture Notes. Volume 36 (2004), 269-285. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 02 2009]
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LINKS
| Casey Mongoven, Phi^2 number 1; electronic music created using Phi^2.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Dmitrij Zelo, Simultaneous Approximation to Real and p-adic Numbers, Feb 28, 2009. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Mar 02 2009]
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FORMULA
| Equals 2+A094214 = 1+A001622. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 19 2008
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EXAMPLE
| 2.618033988...
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MATHEMATICA
| RealDigits[N[GoldenRatio+1, 200]][[1]] (*From Vladimir Joseph Stephan Orlovsky, Feb 20 2011*)
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CROSSREFS
| Cf. A001622.
Sequence in context: A021386 A201936 A019679 * A155832 A136764 A136765
Adjacent sequences: A104454 A104455 A104456 * A104458 A104459 A104460
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KEYWORD
| nonn,cons,easy
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com), Mar 08, 2005
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