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A075778
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Decimal expansion of root of x^3+x^2-1.
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2
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7, 5, 4, 8, 7, 7, 6, 6, 6, 2, 4, 6, 6, 9, 2, 7, 6, 0, 0, 4, 9, 5, 0, 8, 8, 9, 6, 3, 5, 8, 5, 2, 8, 6, 9, 1, 8, 9, 4, 6, 0, 6, 6, 1, 7, 7, 7, 2, 7, 9, 3, 1, 4, 3, 9, 8, 9, 2, 8, 3, 9, 7, 0, 6, 4, 6, 0, 8, 0, 6, 5, 5, 1, 2, 8, 0, 8, 1, 0, 9, 0, 7, 3, 8, 2, 2, 7, 0, 9, 2, 8, 4, 2, 2, 5, 0, 3, 0, 3, 6, 4, 8, 3, 7, 7
(list; constant; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Decimal expansion of an adaptation of the golden Ratio. The square of the lesser is to the greater as the square of the greater is to the whole.
Approximates the sin of 49 degrees and cos 41 degrees nearly forming a right triangle.
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FORMULA
| Let 0 < a < 1 be any real number. Then a is the lesser and 1 is the greater and a^2/1 = 1/(a+1) and a^3 + a^2 - 1 = 0. solving this using PARI we have 0.7548776662466927600495088964... The general cubic can also be solved in radicals.
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EXAMPLE
| Phi golden section = (sqrt(5)-1)/2 = 0.6180339887498948482045868343
x = -(1/3) + (1/3)*(25/2 - (3*Sqrt[69])/2)^(1/3) + (1/3)*((1/2)*(25 + 3*Sqrt[69]))^(1/3)
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MATHEMATICA
| RealDigits[ N[ Solve[x^3 + x^2 - 1 == 0, x] [[1]] [[1, 2]], 111]] [[1]]
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PROG
| (PARI) solve(x=-10, 10, x^3+x^2-1)
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CROSSREFS
| Sequence in context: A154195 A019858 A109134 * A010510 A138313 A138312
Adjacent sequences: A075775 A075776 A075777 * A075779 A075780 A075781
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KEYWORD
| nonn,cons
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Oct 09 2002
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Oct 10 2002
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