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Search: id:A145064
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| A145064 |
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Reduced numerators of the first convergent to the cube root of n using the recursion x = (2*x+n/x^2)/3. |
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+0
1
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| 2, 1, 4, 5, 2, 7, 8, 3, 10, 11, 4, 13, 14, 5, 16, 17, 6, 19, 20, 7, 22, 23, 8, 25, 26, 9, 28, 29, 10, 31, 32, 11, 34, 35, 12, 37, 38, 13, 40, 41, 14, 43, 44, 15, 46, 47, 16, 49, 50, 17, 52, 53, 18, 55, 56, 19, 58, 59, 20, 61, 62, 21, 64, 65, 22, 67, 68, 23, 70, 71, 24, 73, 74, 25
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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The same as A051176 without the first two terms.
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LINKS
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Cino Hilliard, Roots by Recursion
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FORMULA
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The recursion was derived experimentally by analyzing the patterns of root
recursions for polynomials
f(x) = a(n)x^n+a(n-1)x^(n-1)+...+a(1)x+a(0) and
g(x) = a(n-1)x^(n-1)+a(n-2)x^(n-2)+...+a(2)x+a(1)
where the recursion x = a(0)/g(x) may or may not converge to a root and many
iterations are required to get greater accuracy. By introducing an averaging
scheme, a root is found if it exists and convergence is much faster to a root
of f(x) See the link for details. This cubic recursion is equivalent to
Newton's Method.
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PROGRAM
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(PARI) for(k=0, 100, rroot3(k, 1))
rroot3(d, p) = /* Find a root of x^3 - d */Q {
local(x=1, x1=1, j);
for(j=1, p,
x=(x1+x+d/x^2)/3; /* average scheme for a cube root of d */
x1=x; print1(numerator(x)", ");
);
}
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CROSSREFS
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Cf. A051176
Sequence in context: A038502 A106610 A051176 this_sequence A144332 A038719 A125751
Adjacent sequences: A145061 A145062 A145063 this_sequence A145065 A145066 A145067
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KEYWORD
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frac,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)hotmail.com), Sep 30 2008
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